Enhanced pre-ordered pre-weighted transmission

ABSTRACT

Systems and methods for uplink (UL) and downlink (DL) communications can improve channel capacity between one or more access nodes and one or more terminal nodes. The communications may utilize, for example, multi-user multiple-input multiple-output techniques with cooperation between access nodes. A network node may determine an ordering combination and associated pre-weighting values, provide the pre-weighting values to one or more terminal nodes, receive signals transmitted from the terminal nodes using the pre-weighting values; and process the received signals using the ordering combination and the pre-weighting values. Two constraints can be jointly used: a minimum performance constraint and a maximum transmit power constraint. Example systems search all possible ordering combinations to find a best combination, for example, the ordering combination that maximizes communication rates. The ordering combination may be used with combinations of full or partial successive interference cancellation in receivers and full or partial iterative pre-cancellation pre-coding in transmitters.

FIELD OF THE INVENTION

The present invention relates generally to the field of communicationswhere it is desirable to communicate between a number of wirelessdevices at the highest possible communication rate using multipletransmit antennas and multiple receive antennas, while reducing thecomplexity and power consumption of each device.

BACKGROUND

In a wireless communications network, devices intend to communicate withother devices via a communications channel, at the highest possiblerate, with the least possible transmitted power and at the lowestpossible cost. The cost of a device is usually dictated by itscomplexity, while the communication rate between any two devices islimited by Shannon Channel capacity as defined by the communicationmedium between the two devices and by the corresponding transmittedpower. Shannon Channel capacity is a theoretical limit on thecommunication rate, and does not offer an indication on the complexityof the device.

One way to increase Shannon Channel capacity between two devices is toadopt multiple antennas for each device. In the presently disclosedsystems, antennas, which belong to one device, are allowed to cooperate.Another way to increase Shannon Channel capacity is to allow a group ofdevices to transmit simultaneously, to another group of devices. In thepresently disclosed systems and methods, when a group of devices, suchas a transmitting group or a receiving group, is able to cooperate, werefer to it as a cooperative group, otherwise, it is a non-cooperativegroup. For example, in a cellular network, a group of access nodes(e.g., LTE Base Stations) is often a cooperative group, however, a groupof users (subscriber stations, handsets, nodes) is seldom a cooperativegroup.

In this description, we refer to the communication channel from a groupof access nodes to a group of users as a Downlink (DL) Multi-User (MU)channel, while the communication channel from a group of users to agroup of access nodes is referred to as an Uplink (UL) Multi-User (MU)channel. When a device has multiple antennas, we refer to it as aMultiple Input Multiple Output (MIMO) device. When a group of accessnodes with multiple antennas transmit to a group of receiving users alsowith multiple antennas, the network is referred to as a DL MU-MIMOnetwork. On the other hand, when a group of users with multiple antennastransmit to a group of receiving access nodes also with multipleantennas, the network is referred to as a UL MU-MIMO network.

SUMMARY

This disclosure includes systems and methods for UL MU-MIMO and DLMU-MIMO. In both UL and DL MU-MIMO, the presently disclosed systems andmethods can utilize the cooperation between a group of access nodes, toimprove Shannon Channel capacity between access nodes and a group ofusers, under two constraints: a minimum performance constraint per userand a maximum transmit power constraint. More specifically, thepresently disclosed systems and methods can take advantage of thecooperation between a group of access nodes to simultaneously selectpre-weighting values for the transmitted signals and an orderingcombination for the received signals in such a way that the networkcapacity is improved, assuming each access node has some knowledge ofthe communications Channel between access nodes and terminal nodes. Theterm Multi-User (MU) can include single user scenarios since the same orsimilar techniques are generally applicable.

In one aspect, a method for receiving uplink communications in networknode of a communication system is provided. The method includes:determining an ordering combination and determining pre-weighting valuesassociated with the ordering combination for use by at least oneterminal node, the pre-weighting values being determined based on atransmit power constraint and a minimum performance constraint;providing the pre-weighting values to the at least one terminal node;receiving a signal for each of at least one antenna, each receivedsignal comprising a plurality of transmitted signals from the at leastone terminal node based at least in part on the pre-weighting values;and processing the received signal for each of the at least one antennausing the determined ordering combination and the determinedpre-weighting values.

In one aspect, an access node is provided. The access node includes: atransceiver module configured to communicate with at least one terminalnode including receiving a signal via each of at least one antenna, eachreceived signal comprising a plurality of signals transmitted from theat least one terminal node based at least in part on pre-weightingvalues; and a processor module coupled to the transceiver module andconfigured to determine an ordering combination for use in processing atleast one of the received signals and determine pre-weighting valuesassociated with the ordering combination for use by the at least oneterminal node, the pre-weighting values being determined based on atransmit power constraint and a minimum performance constraint; providethe pre-weighting values to the transceiver module for communication tothe at least one terminal node; and process the at least one of thereceived signals using the determined ordering combination and thedetermined pre-weighting values.

BRIEF DESCRIPTION OF THE DRAWINGS

The details of the present invention, both as to its structure andoperation, may be gleaned in part by studying the accompanying drawings,in which like reference numerals refer to like parts, and in which:

FIG. 1 is a block diagram of a communication network in which systemsand methods disclosed herein can be implemented in accordance withaspects of the invention;

FIG. 2A is a block diagram of an access node in accordance with aspectsof the invention;

FIG. 2B is a block diagram of a terminal node in accordance with aspectsof the invention;

FIG. 3 is a block diagram of channels between two transmit antennas(Txs) and two receive antennas (Rxs);

FIG. 4 is a block diagram of multiple transmitter chains in whichaspects of the invention may be implemented;

FIG. 5A is a block diagram of a receiver chain according to an aspect ofthe invention;

FIG. 5B is a block diagram depicting channel estimation, Orderingcombination selection and Pre-weighting calculation according to anaspect of the invention;

FIG. 6 is a flowchart of Ordering combination selection andPre-weighting calculation according to an aspect of the invention;

FIG. 7 is a flowchart of Successive Interference Cancellation (SIC) tosoft-decision estimate and hard-decision detect of received signalsaccording to an aspect of the invention;

FIG. 8 is a flowchart of the application of the set of Pre-weightingvalues to a transmit signal according to an aspect of the invention;

FIG. 9A is a flowchart of an operation for selecting an Orderingcombination and calculating its corresponding Pre-weighting valuesaccording to an aspect of the invention;

FIG. 9B is a flowchart of an operation for selecting an Orderingcombination and calculating its corresponding Pre-weighting valuesaccording to an aspect of the invention;

FIG. 9C is a flowchart of an operation for selecting an Orderingcombination and calculating its corresponding Pre-weighting valuesaccording to an aspect of the invention;

FIG. 9D is a flowchart of an operation for selecting an Orderingcombination and calculating its corresponding Pre-weighting valuesaccording to an aspect of the invention;

FIG. 9E is a flowchart of an operation for selecting an Orderingcombination and calculating its corresponding Pre-weighting valuesaccording to aspect of the invention;

FIG. 9F is a flowchart of an operation to relax the minimum performanceconstraint according to an aspect of the invention;

FIG. 9G is a flowchart of an operation for calculating pre-weightingvalues using relaxed constraints according to an aspect of theinvention;

FIG. 10 is a block diagram of MIMO receiver in which aspects of theinvention may be implemented.

FIG. 11A is a flowchart of Ordering combination for IterativePre-Cancellation (IPC) Pre-coding followed by calculation ofcorresponding Pre-weighting values according to an aspect of theinvention;

FIG. 11B is a flowchart of Ordering combination for Partial IPCPre-coding followed by calculation of corresponding Pre-weighting valuesaccording to an aspect of the invention;

FIG. 11C is a flowchart of Ordering combination for No IPC Pre-codingfollowed by calculation of corresponding Pre-weighting values accordingto an aspect of the invention;

FIG. 12A is a flowchart of the application of the set of Pre-weightingvalues to a transmit signal according to an aspect of the inventionwhich has applied IPC Pre-coding;

FIG. 12B is a flowchart of the application of the set of Pre-weightingvalues to a transmit signal according to an aspect of the inventionwhich has applied Partial IPC Pre-coding; and

FIG. 12C is a flowchart of the application of the set of Pre-weightingvalues according to an aspect of the invention which has No IPCPre-coding.

DETAILED DESCRIPTION

Systems and methods for enhanced pre-weighting transmission areprovided.

FIG. 1 is a block diagram of a communication network in which systemsand methods disclosed herein can be implemented in accordance withaspects of the invention. A macro base station 110 is connected to acore network 102 through a backhaul connection 170. In an embodiment,the backhaul connection 170 is a bidirectional link or twounidirectional links. The direction from the core network 102 to themacro base station 110 is referred to as the downstream or downlink (DL)direction. The direction from the macro base station 110 to the corenetwork 102 is referred to as the upstream or uplink (UL) direction.Users 150(1) and 150(4) can connect to the core network 102 through themacro base station 110. Wireless links 190 between users 150 and themacro base station 110 are bidirectional point-to-multipoint links, inan embodiment. The direction of the wireless links 190 from the macrobase station 110 to the users 150 is referred to as the downlink ordownstream direction. The direction of the wireless links 190 from theusers 150 to the macro base station 110 is referred to as the uplink orupstream direction. Users are sometimes referred to as user equipment(UE), subscriber stations, user devices, handsets, terminal nodes, userterminals, or similar terms and are often mobile devices such as smartphones or tablets. The users 150 access content over the wireless links190 using base stations, such as the macro base station 110, as abridge.

In the network configuration illustrated in FIG. 1, an office building120(1) causes a coverage shadow 104. A pico station 130 can providecoverage to users 150(2) and 150(5) in the coverage shadow 104. The picostation 130 is connected to the core network 102 via a backhaulconnection 170. The users 150(2) and 150(5) may be connected to the picostation 130 via links that are similar to or the same as the wirelesslinks 190 between users 150(1) and 150(4) and the macro base station110.

In office building 120(2), an enterprise femto base station 140 providesin-building coverage to users 150(3) and 150(6). The enterprise femtobase station 140 can connect to the core network 102 via an internetservice provider network 101 by utilizing a broadband connection 160provided by an enterprise gateway 103.

FIG. 2A is a functional block diagram of an access node 275 inaccordance with aspects of the invention. Access nodes may also bereferred to as base stations, Node Bs, access points, base transceiverstations, or similar terms. In various embodiments, the access node 275may be a mobile WiMAX base station, a Universal MobileTelecommunications System (UMTS) NodeB, an LTE evolved Node B (eNB oreNodeB), or other wireless base station or access point of various formfactors. For example, the macro base station 110, the pico station 130,or the enterprise femto base station 140 of FIG. 1 or the macro station111 may be provided, for example, by the access node 275 of FIG. 2. Theaccess node 275 includes a processor module 281. The processor module281 is coupled to a transmitter-receiver (transceiver) module 279, abackhaul interface module 285, and a storage module 283.

The transmitter-receiver module 279 is configured to transmit andreceive communications wirelessly with other devices. The access node275 generally includes one or more antennae for transmission andreception of radio signals. The communications of thetransmitter-receiver module 279 may be with terminal nodes.

The backhaul interface module 285 provides communication between theaccess node 275 and a core network. This may include communicationsdirectly or indirectly (through intermediate devices) with other accessnodes, for example using the LTE X2 interface. The communication may beover a backhaul connection, for example, the backhaul connection 170 ofFIG. 1.

Communications received via the transmitter-receiver module 279 may betransmitted, after processing, on the backhaul connection. Similarly,communication received from the backhaul connection may be transmittedby the transmitter-receiver module 279. Although the access node 275 ofFIG. 2 is shown with a single backhaul interface module 285, otherembodiments of the access node 275 may include multiple backhaulinterface modules. Similarly, the access node 275 may include multipletransmitter-receiver modules. The multiple backhaul interface modulesand transmitter-receiver modules may operate according to differentprotocols. Communications originating within the access node 275, suchas communications with other access nodes, may be transmitted on one ormore backhaul connections by backhaul interface module 285. Similarly,communications destined for access node 275 may be received from one ormore backhaul connections via backhaul interface module 285.

The processor module 281 can process communications being received andtransmitted by the access node 275. The storage module 283 stores datafor use by the processor module 281. The storage module 283 may also beused to store computer readable instructions for execution by theprocessor module 281. The computer-readable instructions can be used bythe access node 275 for accomplishing the various functions of theaccess node 275. In an embodiment, the storage module 283 or parts ofthe storage module 283 may be considered a non-transitorymachine-readable medium. For concise explanation, the access node 275 orembodiments of it are described as having certain functionality. It willbe appreciated that in some embodiments, this functionality isaccomplished by the processor module 281 in conjunction with the storagemodule 283, transmitter-receiver module 279, and backhaul interfacemodule 285. Furthermore, in addition to executing instructions, theprocessor module 281 may include specific purpose hardware to accomplishsome functions.

FIG. 2B is a functional block diagram of a terminal node 255 inaccordance with aspects of the invention. In various embodiments, theterminal node 255 may be a mobile WiMAX user, a UMTS cellular phone, anLTE user equipment, or other wireless terminal node of various formfactors. The users 150 of FIG. 1 may be provided, for example, by theterminal node 255 of FIG. 2B. The terminal node 255 includes a processormodule 261. The processor module 261 is coupled to atransmitter-receiver module (transceiver) 259, a user interface module265, and a storage module 263.

The transmitter-receiver module 259 is configured to transmit andreceive communications with other devices. For example, thetransmitter-receiver module 259 may communicate with the access node 275of FIG. 2A via its transmitter-receiver module 279. The terminal node255 generally includes one or more antennae for transmission andreception of radio signals. Although the terminal node 255 of FIG. 2B isshown with a single transmitter-receiver module 259, other embodimentsof the terminal node 255 may include multiple transmitter-receivermodules. The multiple transmitter-receiver modules may operate accordingto different protocols.

The terminal node 255, in many embodiments, provides data to andreceives data from a person (user). Accordingly, the terminal node 255includes the user interface module 265. The user interface module 265includes modules for communicating with a person. The user interfacemodule 265, in an embodiment, includes a speaker and a microphone forvoice communications with the user, a screen for providing visualinformation to the user, and a keypad for accepting alphanumericcommands and data from the user. In some embodiments, a touch screen maybe used in place of or in combination with the keypad to allow graphicalinputs in addition to alphanumeric inputs. In an alternative embodiment,the user interface module 265 includes a computer interface, forexample, a universal serial bus (USB) interface, to interface theterminal node 255 to a computer. For example, the terminal node 255 maybe in the form of a dongle that can be connected to a notebook computervia the user interface module 265. The combination of computer anddongle may also be considered a terminal node. The user interface module265 may have other configurations and include functions such asvibrators, cameras, and lights.

The processor module 261 can process communications being received andtransmitted by the terminal node 255. The processor module 261 can alsoprocess inputs from and outputs to the user interface module 265. Thestorage module 263 stores data for use by the processor module 261. Thestorage module 263 may also be used to store computer readableinstructions for execution by the processor module 261. Thecomputer-readable instructions can be used by the terminal node 255 foraccomplishing the various functions of the terminal node 255. In anembodiment, the storage module 263 or parts of the storage module 263may be considered a non-transitory machine-readable medium. For conciseexplanation, the terminal node 255 or embodiments of it are described ashaving certain functionality. It will be appreciated that in someembodiments, this functionality is accomplished by the processor module261 in conjunction with the storage module 263, the transmitter-receivermodule 259, and the user interface module 265. Furthermore, in additionto executing instructions, the processor module 261 may include specificpurpose hardware to accomplish some functions. Multiple transmissions ofindependent data streams by using coinciding time-frequency (T/F)resource allocation have been enabled by developments in communicationsystems. These techniques are a subset of a family of techniques thatare called Multiple Input Multiple Output (MIMO) techniques. In MIMOsystems more than one antenna at either or both of the receiver andtransmitter are used. In a specific class of MIMO techniques calledspatial multiplexing, multiple distinct transmissions are resolved fromeach other through using multiple antennas and associated receiverchains at the receiver. In MIMO spatial multiplexing (MIMO-SM), thetransmission data rate is increased by making multiple transmissions atcoinciding T/F resources while using multiple transmission and receptionantennas. Distinct groups of data to be transmitted are referred to aslayers.

Other MIMO techniques include transmitter diversity and receiverdiversity. In transmitter diversity, the same information is eitherdirectly or in some coded form transmitted over multiple antennas. Inreceiver diversity, multiple receive antennas are used to increase thereceived signal quality. Any two or all of the techniques of transmitterdiversity, receiver diversity, and spatial multiplexing (SM) may be usedsimultaneously in a system. For example, a MIMO-SM system may alsodeploy transmitter diversity in addition to receive diversity.

A pre-coding operation in a MIMO transmitter may be used to transmitpre-weighted combinations of signals associated with each of thetransmitted layers by each antenna. Alternately, the pre-codingoperation may be designed such that each of the transmit antennas isused for transmitting a unique layer, for example in the LTE standardfor uplink. The pre-coding operation may map the layers to antennas insuch a manner that the number of transmit antennas is greater that thenumber of transmitted layers.

MIMO-SM techniques include single-user (SU) MIMO-SM and multi-user (MU)MIMO-SM techniques. In SU-MIMO-SM, multiple layers are transmitted by atransmitter at coinciding T/F resources and received by a receiver. InMU-MIMO-SM, multiple signals using common T/F resources are eithertransmitted by multiple transmitters and received by a receiver (e.g.uplink transmission in a cellular network), or, transmitted by a singletransmitter and received by multiple receivers (e.g. Downlinktransmission in a cellular network). We refer to MIMO-SM simply as MIMOand focus generally on MU-MIMO since SU-MIMO forms a subset of MU-MIMO.FIG. 3 provides an example of communication channel impact for the caseof two transmit antennas (Txs) 305 and 307 and two receive antennas(Rxs) 315 and 317 arranged in a MIMO configuration. Identifiers such as“transmitter,” “receiver,” and the like may be omitted from some termswhen the usage context provides such identification.

In the case where transmit antennae (Txs) 305 and 307 are located in asingle transmitter (e.g., an access node such as an LTE eNB basestation) and where receive antennae (Rxs) 315 and 317 are located in asingle receiver (e.g., a terminal node such as a smartphone), then thesystem is in a SU-MIMO configuration. In a SU-MIMO configuration,operation of the multiple transmitter chains in the transmitter may becoordinated, as depicted by the dashed arrow line between transmitterchains 301 and 303. For example, coordination may include support for apre-coding operation.

Similarly, in a SU-MIMO configuration, operation of the multiplereceiver chains in the receiver may be coordinated, as depicted by thedashed arrow line between receiver chains 311 and 313. For example,coordination may include support for a joint decoding operation.

In the case where either transmit antennae 305 and 307 and/or receiveantennae 315 and 317 are located in more than one transmitter orreceiver, respectively, then the system is in a MU-MIMO configuration.

One skilled in the art would appreciate that the use of MU-MIMO andSU-MIMO are not exclusive and that both modes of operation may be usedconcurrently in a multi-user access network such as the communicationnetwork depicted in FIG. 1. Similarly, one skilled in the art wouldappreciate that the various MIMO modes of operation may be applied inboth UL and DL directions without loss of generality.

As there are multiple transmissions that use coinciding T/F resources,each of the receive antennas 315 and 317 are exposed to versions ofsignals from all transmit antennas 305 and 307 each impacted by thechannel transfer function (CTF), which is represented by a matrix,

${h_{Ch}\overset{\Delta}{=}\begin{bmatrix}h_{11}^{ch} & h_{21}^{ch} \\h_{12}^{ch} & h_{22}^{ch}\end{bmatrix}},$

between a particular transmit antenna (Tx) and a particular receiveantenna (Rx). The represented CTF values h₁₁ ^(ch), h₁₂ ^(ch), h₂₁^(ch), h₂₂ ^(ch) depicted in FIG. 3 refer to complex valued CTF valuesbetween the specific antennas per each resource element (e.g., onesubcarrier for one QAM symbol in an OFDM system) of the transmission. Inpractice, each of the transmitter chains 301 and 303 and the receiverchains 311 and 313 are not ideal and may impact the received value foreach transmitted resource element. The techniques provided in thisapplication while referring to the CTF between particular transmit andreceive antennas represented by CTF values h₁₁ ^(ch), h₁₂ ^(ch), h₂₁^(ch), h₂₂ ^(ch) apply with no limitation when the CTF values alsoincorporate the impact of transmitter chains and the receiver chains.Practical real life measurement of CTFs would also reflect the impact ofthe transmitter chains and the receiver chains on each resource element.

In a frequency selective channel, the CTF values may be different fordifferent subcarriers of a received OFDM symbol. In contrast, in afrequency flat channel all CTF values across the frequency range aresubstantially the same within a tolerance (e.g., the magnitude of theCTF values are within some fraction of a decibel of each other). In atime varying channel the CTF values may vary from one OFDM symbol toanother at a same frequency subcarrier. In contrast, in anon-time-varying channel, CTF values do not vary by more than a certainamount from one OFDM symbol to another at a same frequency subcarrier(e.g., the magnitude of the CTF values are within some fraction of adecibel of each other).

FIG. 4 is a block diagram of transmitter chains 401 and 402 in whichaspects of the invention may be implemented. Transmitter chains 401, 402start with the input of the data stream (stream A/stream B) into encoder410 and 420, the output of which is then passed through scrambler 411and 421, modulation mapper 412 and 422, layer mapper 413 and 423,FFT/DFT 414 and 424 (optional, and used for single carrier, frequencydivision multiple access (SC-FDMA) transmissions such as an LTE ULtransmission), pre-weight module 415 and 425, pre-coder 416 and 426,resource mapper 417 and 427, IFFT/IDFT 418 and 428, and then transmittedthrough the antenna 450 and 451. The layer mappers 413, 423 map theencoded and scrambled data streams to MIMO layers. The mapping is doneaccording a particular SM or diversity coding used for transmission.Many of the components of each of transmitter chains 401, 402 are knownto those skilled in the art and are not described in detail herein forthe sake of brevity. As seen in FIG. 4, a set of pre-weighting values419, 429 is introduced into the pre-weighting module 415 and 425 inorder to pre-weight the transmissions of stream A and stream B. FIG. 5Ais a block diagram of a receiver chain 500 according to an exampleembodiment of the present invention. Many of the components of receiverchain 500 are known to those skilled in the art and are not described indetail herein for the sake of brevity. Of particular importance is theFFT 530 which transforms the digital baseband signal into the frequencydomain with FFT outputs 585 for each subcarrier/OFDM symbol pair. Fromthe FFT outputs 585, the resource demapper 535 extracts the referencesignal 580 and data elements 595 from each transmission. Referencesignals 580 are passed to the ordering and pre-weight determiner 590,which is described in more detail with respect to FIG. 5B. Theinformation in the subcarriers corresponding to a transmission'sreference signal is used to create a channel transfer function estimatefor the transmission. In receiver chain 500 of FIG. 5A, IFFT/IDFTdecoder 545 is an optional function and is used for processing thereception of certain transmission waveforms, such as SC-FDMAtransmissions waveforms. In receiver chain 500 of FIG. 5A, a codewordmapper 550 is followed by an SIC detector 555, a descrambler 560 anddecoder 565 to generate the detected data 570. In an aspect, forexample, the receiver portion of transmitter-receiver (transceiver)module 279 of access node 275, may be implemented using receiver chain500.

FIG. 5B is a block diagram describing ordering and pre-weight determiner590, which includes a channel estimation module 592 that acceptsreference signals (580) from one or more receiver chains and generatesan estimate of the channel 594, followed by an ordering module 596 forselecting an ordering combination 598 to be used for the SIC detector atthe receiving devices and a pre-weighting module 582 for calculatingpre-weighting values which correspond to the estimated channel and tothe selected ordering combination, resulting in an ordering combinationand pre-weighting values 584. Ordering combination selection performedby ordering module 596 and pre-weighting calculation performed bypre-weighting module 582 are according to an aspect of the invention.Additionally, the modules of FIG. 5B may obtain information regardingthe group of transmitting devices that transmit signals to a group ofreceiving devices. As used herein, the terms “transmitting device” and“receiving device” are used interchangeably with the terms “transmitter”and “receiver,” respectively. Such information may include one moreinformation elements, and the information about the group oftransmitting devices and the group of receiving devices may be obtained,for example, from a media access control layer scheduler. The term“information element” is used throughout this description in a broadsense to refer to any information that may be communicated between nodesincluding, for example, user data, control signaling, and referencesignals.

FIG. 6 is a flowchart of selecting an ordering combination of theinformation elements and calculating its corresponding pre-weightingvalues according to an aspect of the invention. In FIG. 6, step 601obtains information regarding the group of transmitting devices thattransmit signals to a group of receiving devices. The informationincludes one or more information elements. The information about thegroup of transmitting devices and the group of receiving devices may beobtained, for example, from a media access control layer scheduler. Thesecond step 603 estimates the channel (CTF) matrix by the receivingdevices. The third step 605 selects an ordering combination. The fourthstep 607 calculates the pre-weighting values, which correspond to theordering combination selected in the previous step 605. Finally, thelast step 609 communicates the selected Ordering combination and thecorresponding calculated pre-weighting values to the transmittingdevices. The functions depicted in FIG. 6 may be performed, for example,by a MIMO receiver such as an LTE base station.

FIG. 7 is a flowchart of the Successive Interference Cancellation (SIC)process 700 that is used to detect the transmitted information inreceived signals at the receiving devices according to an aspect of theinvention. In FIG. 7, the first step 701 obtains information regardingthe group of transmitting devices that transmit signals to a group ofreceiving devices. The second step 703 receives the transmitted signals.The third step 705 selects a “next” signal element in received signalbased on the selected ordering combination from FIG. 6. The fourth step707 filters the selected “next” signal element in the received signal.The fifth step 709 performs a soft-decision estimation on the selected“next” signal element in the received signal. This is followed by thesixth step 711, which performs a hard-decision detection on the selected“next” signal element in the received signal, after removing the effectof pre-weighting. If the hard-decision detection of the selected “next”signal element in the received signal is the last one 713, then theprocess ends, otherwise, the effect of the hard-decision detection ofthe selected “next” signal element in the received signal is removedfrom the received signal in the next step 715, and the number ofremaining signal elements in the received signal is decreased by one717.

FIG. 8 is a flowchart of the application of the pre-weighting valuesthat are obtained in the steps described with respect to FIG. 6 to asignal intended for transmission according to an aspect of theinvention. In FIG. 8, the first step 801 receives the pre-weightingvalues for the signal to be transmitted. In step 803, the signal ispre-weighted according to the pre-weighting values in first step 801. Instep 805, the pre-weighted signal is transmitted.

FIGS. 9A, 9B, 9C, 9D, 9E, and 9F are flowcharts of processes forselecting ordering combinations and for calculating the correspondingpre-weighting values according to aspects of the invention. Each processmay be performed, for example, by an access node such as the access nodeof FIG. 2A. The processes use information about groups of transmittingdevices that will transmit to groups of receiving devices and channelestimates to produce ordering combinations and correspondingpre-weighting values. The information about the groups of transmittingand receiving devices may be obtained, for example, from a media accesscontrol protocol. The channel information may be provided, for example,from the receiving devices based on reference signals from thetransmitting devices. The processes determine the ordering combinationsand corresponding pre-weighting values to improve communications betweentransmitting and receiving devices. Mathematical descriptions of theseprocesses and variations thereof are described below.

The processes determine the ordering combinations and correspondingpre-weighting values subject to a transmit power constraint and areceived signal-to-interference-plus-noise-ratio (SINR) constraint. Thetransmit power constraint may indicate a maximum transmit level and mayinclude constraints on individual transmitters, individual transmittingdevices, and the combined power. In other aspects, the transmit powerconstraint may indicate a transmit power level other than a maximumtransmit power level, such as a mean transmit power level, a mediantransmit power level or other statistical variations of transmit powerlevel. The received SINR constraint indicates a desired minimum SINR atthe receiving devices. The transmit power constraint may be based oncapabilities of the transmitting devices and spectrum regulations. TheSINR constraint may be determined, for example, based on communicationsystem performance requirements. Additionally or alternatively, othermeasures of performance (e.g., signal-to-noise ratio, carrier-to-noiseratio, or carrier-to-interference ratio) may be used as a minimumperformance constraint.

FIG. 9A is a flowchart of an operation for selecting an orderingcombination and for calculating the corresponding pre-weighting valuesaccording to an aspect of the invention. FIG. 9A illustrates a first way(Way 1) of selecting an ordering combination and calculating thecorresponding pre-weighting values.

In FIG. 9A, step 901 obtains information regarding the group oftransmitting devices that transmit signals to a group of receivingdevices. In an aspect, step 901 also determines a set of orderingcombinations including all possible combinations. Alternatively a subsetof all possible combinations may be used. Step 903 obtains channelestimation from the receiving devices. In another aspect, step 903 maydetermine the set of ordering combinations.

Step 905 starts a loop to find one or more ordering combinations, amongthe set of ordering combinations, and their corresponding pre-weightingvalues that satisfy both the transmit Power constraint and the receivedSINR constraint. Step 905 selects a first ordering combination forevaluation. Since the process of FIG. 9A evaluates all orderingcombinations and selects a best ordering combination, the sequence inwhich the ordering combinations are evaluated may be changed withoutaffecting the result. In an alternative aspect, if a combination issatisfactory as soon as it is generated, the process may then break outof the loop.

Step 907 calculates the pre-weighting values which correspond to theselected ordering combination and which satisfy the received SINRconstraint. The calculated pre-weighting values are then tested againstthe transmit power constraint in step 908. If the transmit powerconstraint is satisfied, the process continues to step 909; otherwise,the process continues to step 910. In step 909, the ordering combinationand calculated pre-weighting values are saved for use in subsequentsteps. The process continues from step 909 to step 910.

Step 910 tests whether more ordering combinations exist. If moreordering combinations exist, the process continues to step 906;otherwise, the process continues to step 915. In step 906, a nextordering combination is selected. The process then returns to step 907and then step 908 to determine if more ordering combinations satisfyboth the received SINR constraint and the transmit power constraint.

In step 915, the process tests whether at least one satisfactory(satisfying the transmit power constraint and the received SINRconstraint) ordering combination was found. This may be determined, forexample, by noting whether step 909 has saved one or more orderingcombinations. If at least one satisfactory ordering combination wasfound, the process continues to step 911; otherwise, the processcontinues to step 997.

Step 997 relaxes the received SINR constraint. The received SINRconstraint may be relaxed, for example, by subtracting a small value(e.g., 1 dB) or by multiplying by a value less than one (e.g., 0.794).The process then returns to step 905 to evaluate the set of orderingcombinations using the relaxed SINR constraint. Step 997 operates whenall ordering combinations have been evaluated without finding at leastone ordering combination satisfying the received SINR constraint and thetransmit power constraint. In such cases, the received SINR constraintis relaxed in step 997 and the ordering combinations retested using therelaxed received SINR constraint. Relaxation of the received SINRconstraint in step 997 can be repeated until at least one set ofcalculated pre-weighting values is found that satisfies the relaxedreceived SINR constraint and the transmit power constraint. The processshown in FIG. 9A maintains the power constraint intact and allows thereceived SINR constraint to be relaxed.

Step 911 maximizes the sum rate by selecting optimal ordering andpre-weighting values. Step 913 then communicates the optimizedpre-weighting values to the transmitting devices.

FIG. 9B is a flowchart of another process for selecting an orderingcombination and for calculating the corresponding pre-weighting valuesaccording to another aspect of the invention. The process of FIG. 9B isa variation of the first way (Way 1) of selecting an orderingcombination and calculating the corresponding pre-weighting values andis similar to the process of FIG. 9A with like named steps operating inlike fashion except for described differences. The process of FIG. 9Bdiffers from the process of FIG. 9A in cases where no satisfactoryordering combination is found. In the process of FIG. 9B, the receivedSINR and transmit power constraints are met by eliminating one or morepre-weighing values.

From step 915, when no satisfactory ordering combinations were found,the process continues to step 998. In step 998, the number (which isinitially zero) of eliminated pre-weighing values is incremented.Subsequently in step 907, after the pre-weighting values are calculated,the largest pre-weighting values are removed. The number ofpre-weighting values removed is determined from step 998. Step 998operates when all ordering combinations were evaluated without findingat least one ordering combination satisfying the transmit powerconstraint. In such cases, the ordering combinations are retested withthe maximum pre-weighting values eliminated. The number of eliminatedpre-weighting values can be increased until at least one set ofcalculated pre-weighting values is found that satisfies the transmitpower constraint.

FIG. 9C is a flowchart of a process for selecting an orderingcombination and for calculating the corresponding pre-weighting valuesaccording to another aspect of the invention. The process of FIG. 9C isanother variation of the first way (Way 1) of selecting an orderingcombination and calculating the corresponding pre-weighting values andis similar to the process of FIG. 9A with like named steps operating inlike fashion except for described differences. The process of FIG. 9Cdiffers from the processes of FIGS. 9A and 9B in the processing ofordering combinations that do not satisfy both the received SINRconstraint and the transmit power constraint.

In step 908 of the process of FIG. 9C, if the calculated pre-weightingvalues satisfy the transmit power constraint, the process continues tostep 909; otherwise, the process continues to step 917. In step 917, theprocess calculates relaxed pre-weighting values that satisfy thetransmit power constraint. The relaxed pre-weighting values may becalculated by, for example, eliminating the largest pre-weighting valuesor reducing the magnitude of all of the pre-weighting values. Theprocess continues from step 917 to step 909. In step 909 the orderingcombination and the pre-weighting values from step 907 or step 917 aresaved for use in subsequent steps. The process then continues to step910.

Step 910 tests whether more ordering combinations exist. If moreordering combinations exist, the process continues to step 906;otherwise, the process continues to step 911. In step 906, a nextordering combination is selected. The process then returns to step 907to evaluate the next ordering combination.

FIG. 9D is a flowchart of an operation using a second way for selectingan ordering combination and for calculating the correspondingpre-weighting values according to another aspect of the invention. Theprocess (Way 2) of FIG. 9D is similar to the process of FIG. 9C withlike named steps operating in like fashion except for describeddifferences. The process of FIG. 9D calculates pre-weighting values thatsatisfy the transmit power constraint and then tests whether thosepre-weighting values satisfy the received SINR constraint. Accordingly,step 957 calculates pre-weighting values which correspond to theselected ordering combination and which satisfy the transmit Powerconstraint. The calculated pre-weighting values are then tested againstthe SINR constraint in step 958. If the received SINR constraint issatisfied, the process continues to step 909 (where evaluation resultsare saved); otherwise, the process continues to step 937 where relaxedpre-weighting values are calculated.

FIG. 9E is a flowchart of an operation using a third way for selectingan ordering combination and for calculating the correspondingpre-weighting values according to a third aspect of the invention. Theprocess (Way 3) of FIG. 9E is similar to the process of FIG. 9C withlike named steps operating in like fashion except for describeddifferences. The process of FIG. 9E works to calculate pre-weightingvalues that satisfy the transmit power constraint and the received SINRconstraint in one step. Step 978 (reached from step 905 or 906 whichselect the next ordering combination) calculates the pre-weightingvalues which correspond to the selected ordering combination and whichsatisfy both the received SINR constraint and the transmit powerconstraint. From step 978, if the received SINR constraint and thetransmit power constraint are both satisfied, the process continues tostep 909 (where evaluation results are saved); otherwise, the processcontinues to step 917 where relaxed pre-weighting values are calculated.

FIG. 9F is a flowchart of an operation for relaxing the minimumperformance constraint (e.g., received SINR) according to an aspect ofthe invention. The process of FIG. 9F may be used to implement step 917in the process of FIG. 9C. The first step 919 examines the variance ofthe calculated pre-weighting values. If the variance is larger than apre-specified threshold, the process continues to step 921. Step 921finds the maximum value among all calculated pre-weighting values. Thisvalue is removed in step 923 from the set of calculated pre-weightingvalues, and the power constraint is tested in step 925 with the new setof relaxed pre-weighting values. If the power constraint is satisfiedthe process returns; otherwise, the process returns to step 921 to keepremoving the maximum values of the remaining pre-weighting values, oneby one.

If, in step 919, the variance is smaller than or equal to thepre-specified threshold, the process continues to step 927. Step 927reduces the set of pre-weighting values by a small factor generating anew set of relaxed pre-weighting values. The power constraint is testedin step 931 with the new set of relaxed pre-weighting values. If thepower constraint is satisfied, the process returns; otherwise, theprocess returns to step 927 to further reduce the set of pre-weightingvalues by the small factor, with the process continuing until the powerconstraint is satisfied.

FIG. 9G is a flowchart of an operation for calculating pre-weightingvalues using relaxed constraints (e.g., transmit power) according to anaspect of the invention. The process of FIG. 9G is similar to theprocess of FIG. 9F with like named steps operating in like fashionexcept for described differences. The process of FIG. 9G may be used toimplement step 937 in the process of FIG. 9D. The first step 919examines the variance of the calculated pre-weighting values. If thevariance is larger than a pre-specified threshold, the process continuesto step 921. Step 921 finds the maximum value among all calculatedpre-weighting values. This pre-weighting value is removed in step 983from the set of calculated pre-weighting values. The remaining weightingvalues are increased by a corresponding amount. In step 985, the SINRconstraint is tested with the new set of relaxed pre-weighting values.If the SINR constraint is satisfied, the process returns; otherwise, theprocess returns to step 921 to keep removing the maximum values of theremaining pre-weighting values, one by one.

If, in step 919, the variance is smaller than or equal to thepre-specified threshold, the process continues to step 987. Step 987increases the set of pre-weighting values by a small factor generating anew set of relaxed pre-weighting values. The SINR constraint is testedin step 981 with the new set of relaxed pre-weighting values. If theSINR constraint is satisfied, the process returns; otherwise, theprocess returns to step 987 to further increase the set of pre-weightingvalues by the small factor, with the process continuing until the SINRconstraint is satisfied.

FIG. 10 is a block diagram of MIMO receiver in which aspects of theinvention may be implemented. In FIG. 10, the two receiver chains areoperating in a single receiving device. The operation of the receiverdepicted in FIG. 10 may be viewed as an extension of the operation ofthe SISO receiver depicted in FIG. 5A in which the modules depicted inFIG. 5A have similar functions to like named modules in FIG. 10. Certainfunctions in FIG. 10, such as frequency domain equalizer 1040, codewordmapper 1050 and SIC detector 1055 may process signals received by bothantennas in order to properly reconstruct data streams A and B, asunderstood by one skilled in the art.

FIG. 11A is a flowchart for operation of an Iterative Pre-Cancellation(IPC) transmitter. The first step 1101 obtains the transmitter andreceiver grouping information. This is followed by step 1103, whichdetermines the channel information from the receivers. Step 1105 selectsthe ordering combination for IPC pre-coding. This is followed by step1107, which calculates the pre-weighting values corresponding to theselected ordering combination. Step 1105 and step 1107 may be performedusing, for example, the processes of FIGS. 9A-F.

FIG. 11B is a flowchart for operation of a Partial IPC transmitter. Thefirst step 1111 obtains the transmitter and receiver groupinginformation. This is followed by step 1113, which determines the channelinformation from the receivers. Step 1115 selects the orderingcombination for Partial IPC pre-coding. This is followed by step 1117,which calculates the pre-weighting values corresponding to the selectedordering combination. Step 1115 and step 1117 may be performed using,for example, the processes of FIGS. 9A-F.

FIG. 11C is a flowchart for operation of a No IPC transmitter. The firststep 1121 obtains the transmitter and receiver grouping information.This is followed by step 1123, which determines the channel informationfrom the receivers. Step 1125 selects the ordering combination for NoIPC pre-coding. This is followed by step 1127, which calculates thepre-weighting values corresponding to the selected ordering combination.Step 1125 and step 1127 may be performed using, for example, theprocesses of FIGS. 9A-F.

FIG. 12A is a flowchart of the application of the pre-weighting valuesthat are obtained according to the steps shown in FIG. 11A to a signalintended for transmission according to an aspect of the invention. InFIG. 12A, the first step 1201 receives the pre-weighting value, which isapplied to the signal to be transmitted. In the second step 1203, IPCpre-coding is applied to the transmit signal. In the third step 1205,the IPC pre-coded transmit signal is pre-weighted according to thepre-weighting values in first step 1201. In the fourth step 1207, theIPC pre-coded pre-weighted signal is transmitted. Often, the purpose ofapplying IPC pre-coding in step 1203 to the transmit signal in FIG. 12Ais to pre-compensate for the effects of the channel by pre-diagonalizingthe channel matrix. Accordingly, IPC pre-coding may be referred to asprecoding for diagonalization or similar terms and, when used withMU-MIMO, MU-MIMO precoding for diagonalization. It should be appreciatedthat, in other aspects, the steps of FIG. 12A can be implemented in adifferent order. For example, in an aspect, the pre-weighting of step1205 can be performed before the pre-coding of step 1203.

FIG. 12B is a flowchart of the application of the pre-weighting valuesthat are obtained according to the steps shown in FIG. 11B to a signalintended for transmission according to an aspect of the invention. InFIG. 12B, the first step 1211 receives the pre-weighting value, which isapplied to the signal to be transmitted. In the second step 1213,Partial IPC pre-coding is applied to the transmit signal. In the thirdstep 1215, the Partial IPC pre-coded transmit signal is pre-weightedaccording to the pre-weighting values in first step 1211. In the fourthstep 1217, the Partial IPC pre-coded pre-weighted signal is transmitted.Often, the purpose of applying partial IPC pre-coding in step 1213 tothe transmit signal in FIG. 12B is to partially pre-compensate for theeffects of the channel by pre-block diagonalizing the channel matrix.Accordingly, partial IPC pre-coding may be referred to as precoding forblock diagonalization or similar and, when used with MU-MIMO, MU-MIMOprecoding using a block diagonalization function. It should beappreciated that, in other aspects, the steps of FIG. 12B can beimplemented in a different order. For example, in an aspect, thepre-weighting of step 1215 can be performed before the pre-coding ofstep 1213.

FIG. 12C is a flowchart of the application of the pre-weighting valuesthat are obtained according to the steps shown in FIG. 11C to a signalintended for transmission according to an aspect of the invention. InFIG. 12C, the first step 1221 receives the pre-weighting value, whichare applied to the signal to be transmitted. In the second step 1223, NoIPC pre-coding is applied to the transmit signal. In the third step1225, the No IPC pre-coded transmit signal is pre-weighted according tothe pre-weighting values in first step 1221. In the fourth step 1227,the No IPC pre-coded pre-weighted signal is transmitted. Often, thepurpose of applying No IPC pre-coding in step 1223 to the transmitsignal in FIG. 12C is to ignore the effects of the channel by notpre-diagonalizing the channel matrix. It should be appreciated that, inother aspects, the steps of FIG. 12C can be implemented in a differentorder. For example, in an aspect, the pre-weighting of step 1225 can beperformed before the pre-coding of step 1223.

UL MU-MIMO Network:

We now consider an Uplink Multi-User MIMO (UL MU-MIMO) network. Sincethe network is an MU network, we assume that a number, U_(t), oftransmitting devices (e.g., terminal nodes) are assigned to communicatesimultaneously and use overlapping resource elements with a number,U_(r), of receiving devices (e.g., access nodes). Throughout thisdescription, various assumptions are made, for example, to explainaspects or simplify analyses; however, the disclosed systems and methodsmay be still used in applications where the assumptions may not hold.Since the network is a MIMO network, we assume that the v^(th)transmitting device (terminal node) contains a number, N_(t) ^(v), oftransmit antennas and that the w^(th) receiving device (access node)contains a number, N_(r) ^(w), of receive antennas. In other words, theU_(t) transmitting devices (terminal nodes) transmit simultaneouslyN_(t) signal elements at a time (i.e. during one epoch, for example, onesymbol in an LTE system) across a communications channel using a totalof N_(t) transmit antennas (i.e. one signal element per transmitantenna), and the U_(r) receiving devices (access nodes) receivesimultaneously the N_(t) signal elements over a total of N_(r) receiveantennas, where

$N_{t}\overset{\Delta}{=}{{N_{t}^{1} + \ldots + {N_{t}^{U_{t}}\mspace{14mu} {and}\mspace{14mu} N_{r}}}\overset{\Delta}{=}{N_{r}^{1} + \ldots + N_{r}^{U_{r}}}}$

assuming that the propagation time is negligible. We refer to a transmitantenna simply as a Tx and to a receive antenna simply as an Rx.

Equivalently, the U_(t) transmitting devices (terminal nodes) transmitU_(t) pre-weighted signal vectors, {right arrow over (α)}′¹, . . . ,{right arrow over (α)}′^(U) ^(t) (e.g., the signals at transmit antennas305, 307), at a time, i.e. one pre-weighted signal vector pertransmitting device (terminal node) and one pre-weighted signal elementper Tx. The v^(th) transmitting device (terminal node), transmits apre-weighted signal vector, {right arrow over (α)}′^(v), which consistsof N_(t) ^(v) pre-weighted signal elements. The m^(th) element, α′_(m)^(v), in {right arrow over (α)}′^(v) is defined as

$\alpha_{m}^{\prime \; v}\overset{\Delta}{=}{\gamma_{m}^{v}\alpha_{m}^{v}}$

where

-   -   γ_(m) ^(v) is the m^(th) pre-weighting element in {right arrow        over (γ)}^(v);    -   {right arrow over (γ)}^(v)≡N^(v)×1 pre-weighting vector        corresponding to the v^(th) transmitting device (terminal node);        note that “N” as used here is the equivalent of N_(t) as defined        above;    -   α_(m) ^(v) is the m^(th) signal element in a signal vector; and    -   {right arrow over (α)}^(v)≡N^(v)×1 signal vector corresponding        to the v^(th) transmitting device (terminal node); note that “N”        as used here is the equivalent of N_(t) as defined above.

The signal vector {right arrow over (α)}^(v) represents an informationvector,

, which consists of N^(v) elements.

The relationship between the signal vector {right arrow over (α)}^(v),and the information vector,

, can be represented using either a linear function or a non-linearfunction. Examples of such functions include an encoder function, ascrambler function, and an inter-leaver function.

The w^(th) receiving device (access node) receives a received signalvector, {right arrow over (β)}^(w) (e.g., the signals at receiveantennas 315, 317), consisting of N_(r) ^(w) received signal elements,over N_(r) ^(w) Rxs, i.e. one received signal element per Rx. Thecommunications channel between all N_(r) ^(v) Txs (terminal nodeantennas) and the N_(r) ^(w) Rxs (access node antennas) is assumed to beflat fading linear time-invariant (LTI) during one epoch, with additivewhite Gaussian Noise (AWGN), i.e. the channel can be characterized usingthe sub-matrices, h_(Ch) ^(w,1), . . . , h_(Ch) ^(w,U) ^(t) and thenoise can be characterized using a noise vector, {right arrow over(θ)}^(w), during one epoch. When the channel is frequency-selective, itcan be converted to a flat fading channel in the frequency-domain usingcyclic prefix as is done with OFDM. The relationship between {rightarrow over (β)}^(w) and

$\begin{bmatrix}{\overset{\rightarrow}{\alpha}}^{\prime 1} \\\vdots \\{\overset{\rightarrow}{\alpha}}^{\prime \; U_{t}}\end{bmatrix}\overset{\Delta}{=}{\overset{\rightarrow}{\alpha}}^{\prime}$

during one epoch is therefore assumed to be

$\begin{matrix}{{\overset{\rightarrow}{\beta}}^{w} = {{\sum_{v = 1}^{U_{t}}{h_{Ch}^{w,v}{\overset{\rightarrow}{\alpha}}^{\prime \; v}}} + {\overset{\rightarrow}{\theta}}^{w}}} & \left( {1a} \right) \\{= {\begin{matrix}\left\lbrack h_{Ch}^{w,1} \right. & \ldots & {\left. h_{Ch}^{w,U_{t}} \right\rbrack \begin{bmatrix}{\overset{\rightarrow}{\alpha}}^{\prime 1} \\\vdots \\{\overset{\rightarrow}{\alpha}}^{\prime \; U_{t}}\end{bmatrix}}\end{matrix} + {\overset{\rightarrow}{\theta}}^{w}}} & \left( {1b} \right)\end{matrix}$

where

-   -   h_(Ch) ^(w,v)≡N_(r) ^(w)×N_(t) ^(v) is the Channel sub-matrix        between the v^(th) transmitting device (terminal node) and the        w^(th) receiving device (access node), which is defined by its        k^(th) row and m^(th) column element, h_(Ch) _(k,m) ^(w,v);    -   h_(Ch) _(k,m) ^(w,v) represents the complex (time-domain or        frequency-domain) attenuation of the flat fading linear        time-invariant wireless channel connecting the m^(th) transmit        antenna (Tx) of the v^(th) transmitting device to the k^(th)        receive antenna (Rx) of the w^(th) receiving device for        1≦m≦N_(t) ^(v), 1≦k≦N_(r) ^(w), 1≦v≦U_(t), and 1≦w≦U_(r);    -   {right arrow over (α)}′^(v)≡N_(t) ^(v)×1 is the pre-weighted        signal vector for the v^(th) transmitting device, which is        defined by its m^(th) pre-weighted signal element, α′_(m) ^(v);    -   α′_(m) ^(v)        γ_(m) ^(v)α_(m) ^(v) is defined as the pre-weighted signal        element, input to the m^(th) Tx of the v^(th) transmitting        device, where α_(m) ^(v) is the signal element and γ_(m) ^(v) is        the pre-weighting element with 1≦m≦N_(t) ^(v) and 1≦v≦U_(t); in        the transmitter chains of FIG. 4, α_(m) ^(v) may be the outputs        of the FFT/DFT modules 414, 424 and α′_(m) ^(v) may be the input        to the pre-coder modules 416, 426;

${\overset{\rightarrow}{\beta}}^{w}\overset{\Delta}{=}{\begin{bmatrix}\beta_{1}^{w} \\\vdots \\\beta_{N_{r}^{w}}^{w}\end{bmatrix} \equiv {N_{r}^{w} \times 1}}$

is the received signal vector of the w^(th) receiving device (accessnode) using N_(r) ^(w) Rxs, i.e. one element per Rx;

${\overset{\rightarrow}{\alpha}}^{\prime \; v}\overset{\Delta}{=}{\begin{bmatrix}\alpha_{1}^{\prime \; v} \\\vdots \\\alpha_{N_{t}^{v}}^{\prime \; v}\end{bmatrix} \equiv {N_{t}^{v} \times 1}}$

is the pre-weighted signal vector from the v^(th) transmitting device(terminal node) for 1≦v≦U_(t), one element per Tx;

${\overset{\rightarrow}{\alpha}}^{v}\overset{\Delta}{=}{\begin{bmatrix}\alpha_{1}^{v} \\\vdots \\\alpha_{N_{t}^{v}}^{v}\end{bmatrix} \equiv {N_{t}^{v} \times 1}}$

is the signal vector consisting of N_(t) ^(v) signal elements intendedto be transmitted by the v^(th) transmitting device (terminal node)using N_(t) ^(v) Txs, i.e. one signal element per Tx;

$\overset{\rightarrow}{\varsigma}\overset{\Delta}{=}{\begin{bmatrix}\varsigma_{1} \\\vdots \\\varsigma_{N}\end{bmatrix} \equiv {N \times 1}}$

is the information vector consisting of N information elements;

${\overset{\rightarrow}{\gamma}}^{v}\overset{\Delta}{=}{\begin{bmatrix}\gamma_{1}^{v} \\\vdots \\\gamma_{N_{t}^{v}}^{v}\end{bmatrix} \equiv {N_{t}^{v} \times 1}}$

is the pre-weighting vector for the v^(th) transmitting device (terminalnode);

${\overset{\rightarrow}{\theta}}^{w}\overset{\Delta}{=}{\begin{bmatrix}\theta_{1}^{w} \\\vdots \\\theta_{N_{r}^{w}}^{w}\end{bmatrix} \equiv {N_{r}^{w} \times 1}}$

is the noise contaminating the output of the w^(th) receiving device(access node) over the N_(r) ^(w) Rxs, i.e. one noise element per Rx;

-   -   N_(t) ^(v) is the number of Txs in the v^(th) transmitting        device (terminal node) for 1≦v≦U_(t);    -   N_(r) ^(w) is the number of Rxs in the w^(th) receiving device        (access node) for 1≦w≦U_(r); and

$N\overset{\Delta}{=}{N^{1} + \ldots + N^{U_{t}}}$

is the number of information elements in the information vector {rightarrow over (ζ)}

Equation (1b) can be re-written as follows:

$\begin{matrix}{{\overset{\rightarrow}{\beta}}^{w} = {{h_{Ch}^{w}\begin{bmatrix}{\overset{\rightarrow}{\alpha}}^{\prime 1} \\\vdots \\{\overset{\rightarrow}{\alpha}}^{\prime \; U_{t}}\end{bmatrix}} + {\overset{\rightarrow}{\theta}}^{w}}} & \left( {1c} \right)\end{matrix}$

where

$h_{Ch}^{w}\overset{\Delta}{=}{\begin{matrix}\left\lbrack h_{Ch}^{w,1} \right. & \ldots & \left. h_{Ch}^{w,U_{t}} \right\rbrack\end{matrix} \equiv {N_{r}^{w} \times N_{t}}}$

represents the communications Channel over which all U_(t) transmittingdevices (terminal nodes) transmit via their N_(t) Txs to the w^(th)receiving device (access node); and

$N_{t}\overset{\Delta}{=}{N_{t}^{1} + \ldots + N_{t}^{U_{t}}}$

is the total number of Txs in the network.

Equations (1c) can be re-written to include the entire communicationsnetwork, i.e. to include all U_(r) receiving devices with all theirN_(r) Rxs as follows:

$\begin{matrix}\begin{matrix}{\begin{bmatrix}{\overset{\rightarrow}{\beta}}^{1} \\\vdots \\{\overset{\rightarrow}{\beta}}^{U_{r}}\end{bmatrix} = {{\begin{bmatrix}h_{Ch}^{1,1} & \ldots & h_{Ch}^{1,U_{t}} \\\vdots & \ddots & \vdots \\h_{Ch}^{U_{r,}1} & \ldots & h_{Ch}^{U_{r},U_{t}}\end{bmatrix}\begin{bmatrix}{\overset{\rightarrow}{\alpha}}^{\prime 1} \\\vdots \\{\overset{\rightarrow}{\alpha}}^{\prime \; U_{t}}\end{bmatrix}} + \begin{bmatrix}{\overset{\rightarrow}{\theta}}^{1} \\\vdots \\{\overset{\rightarrow}{\theta}}^{U_{r}}\end{bmatrix}}} \\{= {{h_{Ch}\begin{bmatrix}{\overset{\rightarrow}{\alpha}}^{\prime 1} \\\vdots \\{\overset{\rightarrow}{\alpha}}^{\prime \; U_{t}}\end{bmatrix}} + \begin{bmatrix}{\overset{\rightarrow}{\theta}}^{1} \\\vdots \\{\overset{\rightarrow}{\theta}}^{U_{r}}\end{bmatrix}}}\end{matrix} & \left( {2a} \right)\end{matrix}$

where

$h_{Ch}\overset{\Delta}{=}{\begin{bmatrix}h_{Ch}^{1} \\\vdots \\h_{Ch}^{U_{r}}\end{bmatrix} = {\begin{bmatrix}h_{Ch}^{1,1} & \ldots & h_{Ch}^{1,U_{t}} \\\vdots & \ddots & \vdots \\h_{Ch}^{U_{r},1} & \ldots & h_{Ch}^{U_{r},U_{t}}\end{bmatrix} \equiv {N_{r} \times N_{t}}}}$

is referred to as the Channel matrix defined by its sub-matrix, h_(Ch)^(w,v), which is located at the w^(th) column block and at the v^(th)row block of h_(Ch);

-   -   sub-matrix h_(Ch) ^(w,v) connects the v^(th) transmitting device        (terminal node) to the w^(th) receiving device (access node) via        the various Txs in the v^(th) transmitting device (terminal        node) and the various Rxs in the w^(th) receiving device (access        node), for 1≦v≦U_(t), and 1≦w≦U_(r);    -   N_(r) ^(w) is the total number of received signal elements in        {right arrow over (β)}^(w), at the w^(th) receiving device        (access node) for 1≦w≦U_(r);    -   N_(t) ^(v) is the total number of pre-weighted signal elements,        {right arrow over (α)}′^(v), transmitted at the v^(th)        transmitting device (user) for 1≦v≦U_(t);    -   N_(r) is the total number of received signal elements

$\begin{bmatrix}{\overset{\rightarrow}{\beta}}^{1} \\\vdots \\{\overset{\rightarrow}{\beta}}^{U_{r}}\end{bmatrix}\overset{\Delta}{=}\overset{\rightarrow}{\beta}$

across all U_(r) receiving devices (access nodes); and

-   -   N_(t) is the total number of transmitted pre-weighted signal        elements

$\begin{bmatrix}{\overset{\rightarrow}{\alpha}}^{\prime 1} \\\vdots \\{\overset{\rightarrow}{\alpha}}^{\prime \; U_{t}}\end{bmatrix}\overset{\Delta}{=}{\overset{\rightarrow}{\alpha}}^{\prime}$

across all U_(t) transmitting devices (terminal nodes).

Equation (2a) can be re-written in a simpler form as follows

$\begin{matrix}{\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix} = {{h_{Ch}\begin{bmatrix}\alpha_{1}^{\prime} \\\vdots \\\alpha_{N_{t}}^{\prime}\end{bmatrix}} + \begin{bmatrix}\theta_{1} \\\vdots \\\theta_{N_{r}}\end{bmatrix}}} & \left( {2\; b} \right) \\{= {{h_{Ch}\begin{bmatrix}{\alpha_{1}\gamma_{1}} \\\vdots \\{\alpha_{N_{t}}\gamma_{N_{t}}}\end{bmatrix}} + \begin{bmatrix}\theta_{1} \\\vdots \\\theta_{N_{r}}\end{bmatrix}}} & \left( {2c} \right)\end{matrix}$

where

$\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix}\overset{\Delta}{=}{\begin{bmatrix}{\overset{\rightarrow}{\beta}}^{1} \\\vdots \\{\overset{\rightarrow}{\beta}}^{U_{r}}\end{bmatrix}\overset{\Delta}{=}\overset{\rightarrow}{\beta}}$

where the U₄ vectors,

$\begin{bmatrix}{\overset{\rightarrow}{\beta}}^{1} \\\vdots \\{\overset{\rightarrow}{\beta}}^{U_{r}}\end{bmatrix},$

are converted into N_(r) elements,

$\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix};$

$\begin{bmatrix}\alpha_{1}^{\prime} \\\vdots \\\alpha_{N_{t}}^{\prime}\end{bmatrix}\overset{\Delta}{=}{\begin{bmatrix}{\overset{\rightarrow}{\alpha}}^{\prime 1} \\\vdots \\{\overset{\rightarrow}{\alpha}}^{\prime \; U_{t}}\end{bmatrix}\overset{\Delta}{=}{\overset{\rightarrow}{\alpha}}^{\prime}}$

where the U_(t) vectors,

$\begin{bmatrix}{\overset{\rightarrow}{\alpha}}^{\prime 1} \\\vdots \\{\overset{\rightarrow}{\alpha}}^{{\prime U}_{t}}\end{bmatrix},$

are converted into N_(t) elements,

$\quad{\begin{bmatrix}\alpha_{1}^{\prime} \\\vdots \\\alpha_{N_{t}}^{\prime}\end{bmatrix};}$

and

$\begin{bmatrix}\theta_{1} \\\vdots \\\theta_{N_{r}}\end{bmatrix}\overset{\Delta}{=}{\begin{bmatrix}{\overset{\rightarrow}{\theta}}^{1} \\\vdots \\{\overset{\rightarrow}{\theta}}^{U_{r}}\end{bmatrix}\overset{\Delta}{=}\overset{\rightarrow}{\theta}}$

where the U_(r) vectors,

$\quad{\begin{bmatrix}{\overset{\rightarrow}{\theta}}^{1} \\\vdots \\{\overset{\rightarrow}{\theta}}^{U_{r}}\end{bmatrix},}$

are converted into N_(r) elements,

$\begin{bmatrix}\theta_{1} \\\vdots \\\theta_{N_{r}}\end{bmatrix}.$

Similarly, we can convert the U_(t) vectors,

$\begin{bmatrix}{\overset{\rightarrow}{\alpha}}^{1} \\\vdots \\{\overset{\rightarrow}{\alpha}}^{U_{t}}\end{bmatrix},$

into N_(t) elements,

${\begin{bmatrix}\alpha_{1} \\\vdots \\\alpha_{Nt}\end{bmatrix}\overset{\Delta}{=}\overset{\rightarrow}{\alpha}},$

and the U_(t) vectors,

$\begin{bmatrix}{\overset{\rightarrow}{\gamma}}^{1} \\\vdots \\{\overset{\rightarrow}{\gamma}}^{U_{t}}\end{bmatrix},$

into N_(t) elements,

$\begin{bmatrix}\gamma_{1} \\\vdots \\\gamma_{Nt}\end{bmatrix}\overset{\Delta}{=}{\overset{\rightarrow}{\gamma}.}$

The presently disclosed systems and methods work to improve the overallperformance of the communication system represented by Equation (2c) byselecting the N_(t) pre-weighting elements,

$\begin{bmatrix}\gamma_{1} \\\vdots \\\gamma_{N_{t}}\end{bmatrix},$

corresponding to the N_(t) signal elements

$\quad\begin{bmatrix}\alpha_{1} \\\vdots \\\alpha_{N_{t}}\end{bmatrix}$

communicated between the N_(t) Txs and the N_(r) Rxs during one epoch,under (a) a (maximum) Power constraint defined as

|α₁γ₁|²+ . . . +|α_(N) _(t) γ_(N) _(t) |² ≦P

where P is a pre-specified fixed value and (b) a (minimum) Performanceconstraint for each transmitting device. Improving the overallperformance of a communications system can be accomplished in many ways,such as by increasing its overall bandwidth efficiency or by increasingits overall power efficiency, while maintaining a reasonable overallcomplexity. A compromise between both types of efficiencies is toincrease the sum rate, R₁+ . . . +R_(N) _(t) , of the system where R_(m)is the rate of transmission of the m^(th) signal element α_(m). Assumingthat the communications system consists of N parallel channels, i.e. onechannel per information element, then R_(m) is obtained as

R _(m)=

_(m) log₂(1+η_(m))   (3)

where

-   -   _(m) is the equivalent bandwidth of the m^(th) information        element ζ_(m); and    -   η_(m) is the equivalent received Signal-to-Noise Ratio (SNR)        corresponding to the m^(th) information element ζ_(m) that is        transmitted using N_(t) Txs over the communication channel that        is characterized by the matrix, h_(Ch), and which is received by        N_(r) Rxs as part of the received signal vector

$\quad{\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix}.}$

The sum rate, R₁+ . . . +R_(N), defined by Equation (3), of thecommunications system in Equation (2c), is a theoretical upper bound,also referred to as Shannon Channel Capacity. The Channel matrix,h_(Ch), as well as the Method of reception, that is used by the N_(r)receiving devices, determines whether the bound is reached or not. SomeMethods of reception have a low complexity but are sub-optimal, i.e.their corresponding sum rate as defined by Equation (3) cannot begenerally reached, while other methods can be close to optimal, but havea high complexity. The presently disclosed systems and methods use areasonable-complexity (asymptotically) optimal non-linear Method ofReception, referred to as SIC (e.g., performing using process 700 ofFIG. 7 and by the SIC detector 1055 of FIG. 10).

In summary, the UL MU-MIMO aspects of the presently disclosed systemsand methods intend to increase the sum rate, R₁+ . . . +R_(N), asdefined by Equation (3) for an Uplink (UL) MU-MIMO system by applying(at the transmitting devices) N_(t) selected pre-weighting elements,

$\quad{\begin{bmatrix}\gamma_{1} \\\vdots \\\gamma_{N_{t}}\end{bmatrix},}$

corresponding to the N_(t) signal elements

$\quad\begin{bmatrix}\alpha_{1} \\\vdots \\\alpha_{N_{t}}\end{bmatrix}$

to be transmitted, and by receiving the transmitted signals using SICmethod as described in process 700 of FIG. 7 (and shown in FIG. 10 atSIC module 1055) at the receiving devices. The selection of

$\quad\begin{bmatrix}\gamma_{1} \\\vdots \\\gamma_{N_{t}}\end{bmatrix}$

(e.g., as described with reference to step 607) is made under a(maximum) transmitted Power and a (minimum) Performance (sum rate)constraint. The DL MU-MIMO aspects of the presently disclosed systemsand methods intend to increase the sum rate, R₁+ . . . +R_(N), asdefined by Equation (3) for a Downlink (DL) MU-MIMO system by applying(at the transmitting devices) N_(t) selected pre-weighting elements,

$\quad{\begin{bmatrix}\gamma_{1} \\\vdots \\\gamma_{N_{t}}\end{bmatrix},}$

corresponding to the N_(t) signal elements

$\quad\begin{bmatrix}\alpha_{1} \\\vdots \\\alpha_{N_{t}}\end{bmatrix}$

to be transmitted with Interference Pre-Cancellation (IPC), partial IPCor no IPC, and by receiving the transmitted signals using SIC (e.g., asdescribed with reference to process 700 and SIC detector 1055) partialSIC or no SIC at the receiving devices. Once again, the selection of

$\quad\begin{bmatrix}\gamma_{1} \\\vdots \\\gamma_{N_{t}}\end{bmatrix}$

is made under a (maximum) transmitted Power and a (minimum) Performance(sum rate) constraint. Before showing pre-weighting methods for both ULand DL aspects of the presently disclosed systems and methods, we reviewthe general concepts of Channel estimation in UL MU-MIMO and the variousMethods of reception that are generally available for MU-MIMO, eventhough both concepts should be familiar to a person skilled in the art.

Channel Estimation in UL MU-MIMO:

The full knowledge of the channel matrix, h_(Ch), at a receiving device(access node) is sometimes referred to as Channel Side Full Informationat Receiver (CSFIR) which is defined as estimating at all U_(r)receiving devices (access nodes), the communications channel matrixh_(Ch) between all transmitting devices and all receiving devices. Thisis possible as long as all the U_(r) receiving devices (access nodes)are able to cooperate. As used herein, “all” may generally refer to theset of interest. The partial knowledge of the channel matrix, h_(Ch)^(w), at a receiving device (access node) is sometimes referred to asChannel Side Partial Information at Receiver (CSPIR) which is defined asestimating at the w^(th) receiving device (access nodes), thecommunications network matrix, h_(Ch) ^(w), between all transmittingdevices and the w^(th) receiving device and of the interference that issensed by the w^(th) receiving device.

Methods of Reception in MU-MIMO:

The Method of Reception in a MU-MIMO communications system is the methodof extracting the information elements in the information vector

$\quad{\begin{bmatrix}\varsigma_{1} \\\vdots \\\varsigma_{N}\end{bmatrix},}$

from the received signal vector

$\quad{\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix},}$

based on

-   -   Equation (2c) and    -   the 1:1 function which relates the information vector

$\quad\begin{bmatrix}\varsigma_{1} \\\vdots \\\varsigma_{N}\end{bmatrix}$

to the signal vector

$\quad\begin{bmatrix}\alpha_{1} \\\vdots \\\alpha_{N_{t}}\end{bmatrix}$

via a reverse function. Examples of the 1:1 reverse function include ade-coder, a de-scrambler, and a de-inter-leaver.

The Method of reception is generally divided into two parts. The firstpart includes extracting the signal vector

$\quad\begin{bmatrix}\alpha_{1} \\\vdots \\\alpha_{N_{t}}\end{bmatrix}$

from the received signal vector

$\quad{\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix},}$

while the second part includes extracting the information vector

$\quad\begin{bmatrix}\varsigma_{1} \\\vdots \\\varsigma_{N}\end{bmatrix}$

from the signal vector

$\quad{\begin{bmatrix}\alpha_{1} \\\vdots \\\alpha_{N_{t}}\end{bmatrix}.}$

The first part is a classical linear algebra problem which includessolving for a number of N_(t) unknowns,

$\quad{\begin{bmatrix}\alpha_{1} \\\vdots \\\alpha_{N_{t}}\end{bmatrix},}$

using a set of N_(r) linear equations,

$\quad{\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix}.}$

The fact that the set of equations is contaminated by noise makes thefirst part stochastic, and forces it to be an estimation problem inEquation (2c).

The fact that ζ_(m) can only take one of a finite number of values makesthe second part a detection problem. More specifically, the Method ofreception includes two parts:

-   -   1. an estimation part, which provides a soft-decision        estimation, {circumflex over (α)}′_(m), of α′_(m) where        {circumflex over (α)}′_(m) can take any (complex) value as shown        in Equation (2b); and    -   2. a detection part, which provides a hard-decision detection,        {hacek over (ζ)}_(m), of ζ_(m), based on the soft-decision        estimation, {circumflex over (α)}′_(m) where detected        information element {hacek over (ζ)}_(m) can take only one of a        finite number of (complex) values.

The first part, i.e. the estimation part, of the Method of receptionincludes providing a soft-decision solution,

$\quad{\begin{bmatrix}{\hat{\alpha}}_{1}^{\prime} \\\vdots \\{\hat{\alpha}}_{N_{t}}^{\prime}\end{bmatrix},}$

for the N_(t) unknowns,

$\quad{\begin{bmatrix}{\hat{\alpha}}_{1}^{\prime} \\\vdots \\{\hat{\alpha}}_{N_{t}}^{\prime}\end{bmatrix},}$

in Equation (2b) using the set of N_(r) observed values:

$\quad{\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix}.}$

From a linear algebra point of view and after ignoring the noise, wehave the following three situations:

-   -   When N_(t)<        _(Ch), where        _(Ch) is the rank of h_(Ch), the problem is said to be        over-determined (i.e. we have more equations than unknowns),        which implies that no unique solution exists which satisfies all        equations. However, in this case, a solution can be found as a        compromising fit between all equations (e.g. a Least Squares fit        using a matrix pseudo-inverse).    -   When N_(t)=        _(Ch), the problem is said to be exactly-determined, which        implies that a unique solution exists for all N_(t) unknowns        (using a matrix inverse).    -   When N_(t)>        _(Ch), the problem is said to be under-determined (i.e. we have        more unknowns than independent equations), which implies that an        infinite number of solutions exists. In this case, a solution        can be selected which fits a specific need. For example, a        system may rely on the second part of the method of reception,        i.e. the detection part, since it provides a hard-decision        solution, {hacek over (α)}_(m), based on {hacek over (α)}′_(m).        Since {hacek over (α)}_(m) can only take one of a finite number        of values, one can select the value, {hacek over (α)}_(m), in        the finite set which is closest to {hacek over (α)}′_(m)/γ_(m)        in a Euclidean Distance sense (such as an        ₂-norm), or in any other sense (such as an        ₁-norm or an        _(∞)-norm).

If the method of reception is chosen to be linear, the estimation partgenerally includes a number of linear operations, while the detectionpart simply includes a hard-decision detector.

Examples of a Linear Method of Reception Include:

-   -   Matrix-based Linear Method of Reception which generally includes        two parts:        -   an Estimation part, which includes multiplying the set of            N_(r) observed values,

$\quad{\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix},}$

-   -    by an estimation matrix, h_(Est) (also sometimes known as        filtering or nulling or estimating or equalizing or inverting or        deconvolving); and        -   a Detection part, which includes detecting the N unknowns,

$\quad{\begin{bmatrix}\varsigma_{1} \\\vdots \\\varsigma_{N}\end{bmatrix},}$

using a hard-decision detector.

-   -   Mathematically, the estimation matrix, h_(Est), generates an        estimated (soft-decision) value of the        _(Net) unknowns        from

$\quad\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix}$

as where

$\begin{matrix}{\hat{{\overset{\rightarrow}{\alpha}}_{_{Ch}}^{\prime}} = {h_{Est}\mspace{14mu} {\quad\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix}}}} & \left( {4a} \right)\end{matrix}$

-   -   -   β_(k) is the received signal element that is received at the            k^(th) Rx with 1≦k≦N_(r);        -   h_(Est)≡            _(Ch)×N_(r) estimation matrix which estimates            from

$\quad{\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix};}$

-   -   -   ≡            _(Ch)×1 is the estimated pre-weighted signal vector; and        -   _(Ch) is the rank of the Channel matrix h_(CH).

    -   By substituting

$\quad\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix}$

from Equation (2b) in Equation (4a), we obtain

$\begin{matrix}{\hat{{\overset{\rightarrow}{\alpha}}_{_{Ch}}^{\prime}} = {h_{Est}\mspace{11mu} h_{Ch}\; {\quad{\begin{bmatrix}\alpha_{1}^{\prime} \\\vdots \\\alpha_{N_{t}}^{\prime}\end{bmatrix} + {h_{Est}{\quad\begin{bmatrix}\theta_{1} \\\vdots \\\theta_{N_{r}}\end{bmatrix}}}}}}} & \left( {4b} \right)\end{matrix}$

-   -   Based on the relationship between        _(Ch) and N_(t), we can generally have any one of the following        three problems: an over-determined problem, an        exactly-determined problem and an under-determined problem. The        first two problems have unique solutions for all N_(t) unknowns,        while the third problem provides only unique solutions for up to        _(Ch) unknowns, forcing the remaining N_(t)−        _(Ch) unknowns to be ignored, i.e. to act as interference on the        first        _(Ch) unknowns. That is why the SINR, η_(m), for the m^(th)        information element, α_(m), is used in the presently disclosed        systems and methods in Equation (3) as a signal quality        indicator instead of its Received Signal Strength Indicator        (RSSI) or its Signal-to-Noise Ratio (SNR).

Even though a linear Method of reception is low in complexity, it isoften sub-optimal. A non-linear method of reception can offer asignificant improvement over a linear one, since in this case, one canobtain unique solutions for all N_(t) unknowns instead of only for

_(Ch) unknowns. This is discussed later. First, we show here examples ofthe estimation matrix, h_(Est):

-   -   -   The Matched Filter (MF) estimation matrix,

${h_{MF}\overset{\Delta}{=}h_{Ch}^{*}},$

is designed to match the Channel matrix h_(Ch), where h*_(Ch) is theHermitian of h_(Ch) (conjugate transpose). The MF can be implemented inany domain such as the time domain, the frequency domain or the spatialdomain. The MF estimation matrix, h_(MF), is designed to maximize thereceived SNR. In this case we have

$\begin{matrix}\begin{matrix}{\hat{{\overset{\rightarrow}{\alpha}}_{_{Ch}}^{\prime}} = {h_{Est}\mspace{14mu} {\quad\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix}}}} \\{= {h_{MF}\mspace{11mu} h_{Ch}\; {\quad{\begin{bmatrix}\alpha_{1}^{\prime} \\\vdots \\\alpha_{N_{t}}^{\prime}\end{bmatrix} + {h_{MF}{\quad\begin{bmatrix}\theta_{1} \\\vdots \\\theta_{N_{t}}\end{bmatrix}}}}}}} \\{= {{h_{Ch}^{*}{h_{Ch}\begin{bmatrix}\alpha_{1}^{\prime} \\\vdots \\\alpha_{N_{t}}^{\prime}\end{bmatrix}}} + {h_{Ch}^{*}\begin{bmatrix}\theta_{1} \\\vdots \\\theta_{N_{t}}\end{bmatrix}}}}\end{matrix} & \left( {4c} \right)\end{matrix}$

-   -   The MF is adequate at low SINR. It is sometimes referred to as        Maximum Ratio Combining, or Beam forming.    -   The Zero Forcing (ZF) estimation matrix, h_(ZF), is designed to        satisfy the following identity

h_(ZF) h _(Ch) =I _(N) _(t)   (5)

-   -   where I_(N) _(t) is the N_(t)×N_(t) identity matrix. If h_(Ch)        ⁻¹ exists and N_(t)=        _(Ch) (i.e. exactly-determined), then h_(ZF)=h_(Ch) ⁻¹ satisfies        Equation (5). In this case, we have

$\begin{matrix}{\hat{{\overset{\rightarrow}{\alpha}}_{_{Ch}}^{\prime}} = {{h_{ZF}\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix}} = {{h_{Ch}^{- 1}{h_{Ch}\begin{bmatrix}\alpha_{1}^{\prime} \\\vdots \\\alpha_{N_{t}}^{\prime}\end{bmatrix}}} + {h_{Ch}^{- 1}\begin{bmatrix}\theta_{1} \\\vdots \\\theta_{N_{t}}\end{bmatrix}}}}} & \left( {4d} \right)\end{matrix}$

-   -   On the other hand, when N_(t)<        _(Ch) (i.e. h_(Ch) is not square but rectangular), and        (h*_(Ch)h_(Ch))⁻¹ exists, then h_(ZF) can be implemented to        satisfy Equation (5) as the Moore-Penrose pseudo-inverse matrix,        h_(Ch) ^(pinv), i.e. h_(ZF)=h_(Ch) ^(pinv) which is defined as

$h_{Ch}^{pinv}\overset{\Delta}{=}{\left( {h_{Ch}^{*}h_{Ch}} \right)^{- 1}h_{Ch}^{*}}$

where h*_(Ch) is the Hermitian of h_(Ch). In this case, we have

$\begin{matrix}{{\hat{{\overset{\rightarrow}{\alpha}}^{\prime}}}_{_{Ch}} = {{h_{ZF}\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix}} = {{h_{Ch}^{pinv}{h_{Ch}\begin{bmatrix}\alpha_{1}^{\prime} \\\vdots \\\alpha_{N_{t}}^{\prime}\end{bmatrix}}} + {h_{Ch}^{pinv}\begin{bmatrix}\theta_{1} \\\vdots \\\theta_{N_{t}}\end{bmatrix}}}}} & \left( {4e} \right)\end{matrix}$

-   -   The ZF estimation matrix, h_(ZF), is designed to remove the        effects of the Channel matrix, h_(Ch), up to the channel rank,        _(Ch), regardless of the type of noise contaminating the        received signal. In other words, as long as        _(Ch)≧N_(t), the ZF estimation matrix, h_(ZF), is able to        entirely remove the effects of the Channel matrix, h_(Ch). In        Equations (4d) and (4e), neglecting the effects of the noise can        cause noise enhancement due to multiplying the noise by h_(Ch)        ⁻¹ or by h_(Ch) ^(pinv) respectively. When        _(Ch)<N_(t), (h*_(Ch)h_(Ch))⁻¹ does not exist or hu_(Ch) is        close to being singular, the ZF matrix is said to be        ill-conditioned, and a different matrix must be used. ZF is also        referred to as inverse filtering, or deconvolution.    -   The Minimum Mean Square Error (MMSE) estimation matrix (also        sometimes referred to as the Wiener filter), h_(MMSE), is        defined as

$\begin{matrix}{h_{MMSE}\overset{\Delta}{=}{\left( {{h_{Ch}^{*}h_{Ch}} + {I_{N_{t}}\text{/}\eta}} \right)^{- 1}h_{Ch}^{*}}} & (6)\end{matrix}$

where

-   -   -   I_(N) _(t) ≡N_(t)×N_(t) identity matrix; and        -   η is the received SNR.

    -   The MMSE estimation matrix, h_(MMSE), is designed to minimize        the mean squared error between the received signal and the        transmitted signal. By doing so, it represents a compromise        between noise and interference. At high SINR, it approaches the        Zero Forcing Filter. At low SINR, it approaches the MF. However,        once again, at high SINR, the MMSE estimation matrix, h_(MMSE),        can remove the effects of the Channel matrix, h_(Ch), up to the        channel rank,        _(Ch).

    -   Transform-based Linear method of reception includes three steps:        -   applying the Inverse Transform, ℑ⁻¹,        -   filtering, and        -   detecting using a hard-decision detector.

    -   The Inverse Transform, ℑ⁻¹, is selected to invert the effects of        a transform, ℑ, that is either explicitely used in the        transmitter or implicitly equivalent to multiplexing using a        Channel matrix, h_(Ch), while the filtering is designed to        invert the effects of the Channel matrix. An example of ℑ is the        inverse Fourier transform, used in the OFDM TX. In this case,        ℑ⁻¹ is the Fourier transform. OFDM with cyclic convolution        converts a frequency-selective channel in the time-domain to a        flat fading channel in the frequency-domain. This is why the        presently disclosed systems and methods can be used for either a        flat fading channel or a frequency-fading channel.

Despite the fact that a linear method of reception is simple toimplement, it is inherently sub-optimal in a non-orthogonal system, i.e.in a system where h*_(Ch)h_(Ch) is not diagonal. Its performancedegrades rapidly compared to the optimal solution when the problem isunder-determined or as h_(Ch) gets close to being singular. The reasonbehind this increased degradation is due to the fact that only

_(Ch) unknowns can be detected when the problem is under-determined,forcing the remaining N_(t)−

_(Ch) unknowns to be ignored, i.e. to act as interference on the first

_(Ch) unknowns, selected for detection. Even when the problem isover-determined or exactly-determined, the resulting noise

$h_{Ch}^{- 1}\begin{bmatrix}\theta_{1} \\\vdots \\\theta_{N_{t}}\end{bmatrix}$

in Equation (4d), and

$h_{Ch}^{pinv}\begin{bmatrix}\theta_{1} \\\vdots \\\theta_{N_{t}}\end{bmatrix}$

in Equation (4e), are often enhanced by the estimation matrix, h_(Est),relative to the original noise vector,

$\begin{bmatrix}\theta_{1} \\\vdots \\\theta_{N_{t}}\end{bmatrix}.$

For this reason, non-linear methods of reception must be considered whenthe Channel is non-orthogonal or when

_(Ch)<N_(t).

Examples of Non-Linear Methods of Reception Include:

-   -   the Maximum Likelihood (ML) Method of reception which is optimal        in most cases. It can be implemented using the detection part        only as follows. When the N information elements in

$\quad\begin{bmatrix}\varsigma_{1} \\\vdots \\\varsigma_{N}\end{bmatrix}$

take equally likely values from a finite set of values and when thenoise is AWGN, the ML detects the information vector,

$\begin{bmatrix}\varsigma_{1} \\\vdots \\\varsigma_{N}\end{bmatrix},$

using a hard-decision detector which maximizes the likelihood function,or equivalently it obtains a detected information elements

$\quad\begin{bmatrix}{\overset{ˇ}{\varsigma}}_{1} \\\vdots \\{\overset{ˇ}{\varsigma}}_{N}\end{bmatrix}$

for information elements

$\quad\begin{bmatrix}\varsigma_{1} \\\vdots \\\varsigma_{N}\end{bmatrix}$

as

$\begin{matrix}{\quad{\begin{bmatrix}{\overset{ˇ}{\varsigma}}_{1} \\\vdots \\{\overset{ˇ}{\varsigma}}_{N}\end{bmatrix} = {\arg \left\{ {\min\limits_{{\lbrack\begin{matrix}\varsigma_{1} \\\vdots \\\varsigma_{N}\end{matrix}\rbrack} \in}\left\{ {{{h_{Ch}\begin{bmatrix}{\alpha_{1}\gamma_{1}} \\\vdots \\{\alpha_{N_{t}}\gamma_{N_{t}}}\end{bmatrix}} - \begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix}}}^{2} \right\}} \right\}}}} & (7)\end{matrix}$

where

$\min\limits_{{\lbrack\begin{matrix}\varsigma_{1} \\\vdots \\\varsigma_{N}\end{matrix}\rbrack} \in}$

is performed over the set of finite values,|

|, that the elements of

$\quad\begin{bmatrix}\varsigma_{1} \\\vdots \\\varsigma_{N}\end{bmatrix}$

can take.

Note:

-   -   The complexity of ML grows exponentially with the number of        elements, N_(t), in the information vector

$\quad{\begin{bmatrix}\varsigma_{1} \\\vdots \\\varsigma_{N}\end{bmatrix};}$

-   -   ML requires CSFIR;    -   ML can detect

$\quad\begin{bmatrix}{\overset{ˇ}{\varsigma}}_{1} \\\vdots \\{\overset{ˇ}{\varsigma}}_{N}\end{bmatrix}$

instead of

since it is not limited by the rank,

_(Ch), of the Channel matrix. On the other hand, when using a linearmethod of reception, the number of elements to be soft-decisionestimated, is limited by

_(Ch);

-   -   ML Performance: The performance of ML is upper bounded by the        so-called minimum Euclidean distance d_(min) which is defined as

$\begin{matrix}{{d_{\min}\left( \overset{\rightarrow}{\gamma} \right)} = {\min\limits_{{\lbrack\begin{matrix}\varsigma_{1,1} \\\vdots \\\varsigma_{N,1}\end{matrix}\rbrack} \neq {\lbrack\begin{matrix}\varsigma_{1,2} \\\vdots \\\varsigma_{N,2}\end{matrix}\rbrack}}\left\{ \left| {{h_{Ch}\begin{bmatrix}{\alpha_{1,1}\gamma_{1}} \\\vdots \\{\alpha_{N_{t},1}\gamma_{N_{t}}}\end{bmatrix}} - {h_{Ch}\begin{bmatrix}{\alpha_{1,2}\gamma_{1}} \\\vdots \\{\alpha_{N_{t},2}\gamma_{N_{t}}}\end{bmatrix}}} \right|^{2} \right\}}} & (8)\end{matrix}$

where

${{\bullet \mspace{14mu} \overset{\rightarrow}{\gamma}}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1} \\\vdots \\\gamma_{N_{t}}\end{bmatrix}};$

and

${\bullet \mspace{14mu}\begin{bmatrix}\varsigma_{1,1} \\\vdots \\\varsigma_{N,1}\end{bmatrix}}\mspace{14mu} {{and}\mspace{14mu}\begin{bmatrix}\varsigma_{1,2} \\\vdots \\\varsigma_{N,2}\end{bmatrix}}$

are any two information vectors, which are not identical.

-   -   It is easily shown that in order to increase the sum rate, R₁+ .        . . +R_(N)=Σ_(m=1) ^(N)        _(m) log₂(1+η_(m)) one must select {right arrow over (γ)} in        such a way that d_(min)({right arrow over (γ)}) in Equation (8)        is increased under the constraint that |α₁γ₁|²+ . . . +|α_(N)        _(t) γ_(N) _(t) |²≦P. This is equivalent to solving for

$\begin{matrix}{\hat{\overset{\rightarrow}{\gamma}} = {\arg \left\{ {\max\limits_{\overset{\rightarrow}{\gamma}}\left\{ {d_{\min}\left( \overset{\rightarrow}{\gamma} \right)} \right\}} \right\}}} & (9)\end{matrix}$

under the constraint that |α₁γ₁|²+ . . . +|α_(N) _(t) γ_(N) _(t) |²≦P.That is why {right arrow over (γ)} should be selected to be complex(i.e. to have an amplitude and a phase) in order to be able todistinguish (i.e. increase the Euclidean distance) between the desired

_(Ch) information elements and the interfering N_(t)−

_(Ch) interfering information elements.

-   -   A lower complexity alternative to the ML is the SIC Method of        Reception (e.g., as described with reference to process 700 and        SIC detector 1055), which (asymptotically) converges to the        optimal solution at high SINR and which can detect more than        _(Ch) unknowns. It includes three stages:        -   filtering (soft-decision estimating);        -   hard-decision detecting; and        -   cancelling.    -   SIC is performed using a detection order which may be        determined, for example, from step 605 of FIG. 6. The three        stages are repeated N_(t) times, one iteration per element of        the signal vector

$\begin{bmatrix}\alpha_{1} \\\vdots \\\alpha_{N_{t}}\end{bmatrix}.$

At the i^(th) iteration, the three stages are explained as follows.

-   -   -   the filtering stage (e.g., as described with reference to            step 707) is similar to the filtering stage in the linear            method of reception (i.e. in the linear estimation part)            where filtering can take place either using a filter or by            multiplying using an estimation matrix, in the time-domain            or in the frequency-domain. However, in this case, we are            estimating a single value, α′₁ _(i,o) , (or a group of            values) instead of an entire vector a {right arrow over            (α)}′_(i,o) where α′₁ _(i,o) is the 1^(st) element in {right            arrow over (α)}′_(i,o) corresponding to the information            element, α₁ _(i,o) , of the highest signal quality, i.e. of            the largest received Signal-to-Noise & Interference (SINR),            among all signal elements

$\begin{bmatrix}\alpha_{1_{i,o}} \\\vdots \\\alpha_{N_{t_{i,o}}}\end{bmatrix}.$

Therefore, we use a row vector,

h_(Est_(1_(i, o))),

instead of an entire matrix h_(Est) _(i,o) to obtain the (e.g., fromstep 709) estimated value {circumflex over (α)}′₁ _(i,o) of α′₁ _(i,o)as follows

$\begin{matrix}{{\hat{\alpha}}_{1_{i,o}}^{\prime}\overset{\Delta}{=}{h_{{Est}_{1_{i,o}}}\begin{bmatrix}\beta_{1_{i,o}} \\\vdots \\\beta_{N_{r_{i,o}}}\end{bmatrix}}} & (11)\end{matrix}$

where

-   -   -   -   {circumflex over (α)}′₁ _(i,o) is the estimated value of                the 1^(st) element, α′₁ _(i,o) , in {right arrow over                (α)}′_(i,o);            -   h_(Est) _(1i,o) ≡1×N_(r) is the 1^(st) row vector of the                estimation matrix h_(Est) _(i,o) which is used to                estimate the 1^(st) pre-weighted signal element, α′₁                _(i,o) , in {right arrow over (α)}′_(i,o); and

${\bullet \mspace{14mu}\begin{bmatrix}\beta_{1_{i,o}} \\\vdots \\\beta_{N_{r_{i,o}}}\end{bmatrix}} \equiv {N_{r} \times 1}$

is a vector consisting of the received signal elements which remainafter removing the effects of the (i−1) previously detected informationelements. More specifically,

$\begin{matrix}{\begin{bmatrix}\beta_{1_{i,o}} \\\vdots \\\beta_{N_{r_{i,o}}}\end{bmatrix} = {{{h_{{Ch}_{i,o}}\begin{bmatrix}\alpha_{1_{i,o}}^{\prime} \\\vdots \\\alpha_{N_{t_{i,o}}}^{\prime}\end{bmatrix}} + \begin{bmatrix}\theta_{1_{i,o}} \\\vdots \\\theta_{N_{r_{i,o}}}\end{bmatrix}} = {{h_{{Ch}_{i,o}}\begin{bmatrix}{\alpha_{1_{i,o}}\gamma_{1_{i,o}}} \\\vdots \\{\alpha_{N_{t_{i,o}}}\gamma_{N_{t_{i,o}}}}\end{bmatrix}} + \begin{bmatrix}\theta_{1_{i,o}} \\\vdots \\\theta_{N_{r_{i,o}}}\end{bmatrix}}}} & (12)\end{matrix}$

where

-   -   -   -   h_(Ch) _(i,o) ≡N_(r)×(N_(t)−i+1)is the ordered Channel                sub-matrix which consists of the columns of the Channel                matrix, h_(Ch), corresponding to {right arrow over                (α)}′_(i,o); and

${\bullet \mspace{14mu}\begin{bmatrix}\theta_{1_{i,o}} \\\vdots \\\theta_{N_{r_{i,o}}}\end{bmatrix}} \equiv {N_{r} \times 1}$

is the ordered noise vector.

-   -   -   The choice of h_(Est) _(i,o) is the Matched Filter, the Zero            Forcing Filter, the MMSE filter, etc. . . . with {right            arrow over (γ)} taken into consideration.        -   the detecting stage (e.g., using step 711), which detects            the 1^(st) information element ζ₁ _(i,o) in {right arrow            over (ζ)}_(i,o) by            -   dividing the estimated element {circumflex over (α)}₁                _(i,o) in Equation (11) by γ₁ _(i,o) , then            -   obtaining the estimated information element {circumflex                over (ζ)}₁ _(i,o) of the corresponding information                element, ζ₁ _(i,o) by performing an operation on

${{\hat{\alpha}}_{1_{i,o}}^{\prime}\text{/}\gamma_{1_{i,o}}}\overset{\Delta}{=}{\hat{\alpha}}_{1_{i,o}}$

which is the reverse to the operation which produced α₁ _(i,o) .

-   -   -   -   This is followed by taking a hard-decision detection,                {hacek over (ζ)}₁ _(i,o) , of {circumflex over (ζ)}₁                _(i,o) , as

$\begin{matrix}{{\overset{\bigvee}{\varsigma}}_{1_{i,o}} = {\arg \left\{ {\min\limits_{\mu_{k \in }}\left\{ \left| {{\overset{\bigvee}{\varsigma}}_{1_{i,o}} - \mu_{k}} \right|^{2} \right\}} \right\}}} & (13)\end{matrix}$

where μ_(k) is the k^(th) value in the set of the finite values,

, that ζ_(m) can take; and

-   -   -   the cancelling stage (e.g., as described with reference to            step 715), which removes the effect of detected information            element {hacek over (ζ)}₁ _(i,o) from

$\begin{bmatrix}\beta_{1_{i,o}} \\\vdots \\\beta_{N_{r_{i,o}}}\end{bmatrix}\quad$

by first forming a vector

$\begin{bmatrix}\delta_{1_{i,o}} \\\vdots \\\delta_{N_{r_{i,o}}}\end{bmatrix}\quad$

defined as

$\begin{matrix}{\begin{bmatrix}\delta_{1_{i,o}} \\\vdots \\\delta_{N_{r_{i,o}}}\end{bmatrix}\overset{\Delta}{=}{h_{{Ch}_{i,o}}{\overset{\bigvee}{\alpha}}_{1_{i,o}}\gamma_{1_{i,o}}}} & (14)\end{matrix}$

where detected signal element {hacek over (α)}₁ _(i,o) corresponds todetected information element {hacek over (ζ)}₁ _(i,o) assuming that{hacek over (ζ)}₁ _(i,o) =ζ₁ _(i,o) , i.e. assuming that there is noerror propagation, which is usually true at a high received SINR. Thisis followed by subtracting

$\quad\begin{bmatrix}\delta_{1_{i,o}} \\\vdots \\\delta_{N_{r_{i,o}}}\end{bmatrix}$

from

$\quad\begin{bmatrix}\beta_{1_{i,o}} \\\vdots \\\beta_{N_{r_{i,o}}}\end{bmatrix}$

to form

$\quad\begin{bmatrix}\beta_{1_{{i + 1},o}} \\\vdots \\\beta_{N_{r_{{i + 1},o}}}\end{bmatrix}$

as

$\begin{matrix}{\begin{bmatrix}\beta_{1_{{i + 1},o}} \\\vdots \\\beta_{N_{r_{{i + 1},o}}}\end{bmatrix}\overset{\Delta}{=}{\begin{bmatrix}\beta_{1_{i,0}} \\\vdots \\\beta_{N_{r_{i,o}}}\end{bmatrix} - \begin{bmatrix}\delta_{1_{i,o}} \\\vdots \\\delta_{N_{r_{i,o}}}\end{bmatrix}}} & (15)\end{matrix}$

Method “A” for Selecting Ordering and Pre-Weighting for UL MU-MIMO:

Assumptions “A”:

-   1. Since the network is a UL network, the method assumes that the    U_(r) receiving devices (access nodes) are able to cooperate. This    is a realistic assumption when the receiving devices are access    nodes belonging to the same network. Otherwise, the non-cooperating    access nodes will see each another as interfering networks.-   2. Since the network is a UL network, the method assumes that the    channel matrix, h_(Ch), (or portions of the channel matrix such as    h_(Ch) ^(w)) is known by the U_(r) receiving devices (access nodes).    This is also a realistic assumption since transmissions commonly    include pilot symbols or training sequences, and the receiving    devices typically carry out Channel estimation.-   3. Based on the two previous assumptions that the channel matrix,    h_(Ch), (or portions of the channel matrix such as h_(Ch) ^(w)) is    known by the U_(r) receiving devices and that the U_(r) receiving    devices are able to cooperate, the method assumes that SIC is    selected as the Method of reception at the U_(r) receiving devices    (access nodes) after filtering the received signals. This assumption    can be realistically carried out by implementing the three stages of    SIC: Filtering, Hard-decision detecting and Cancelling.-   4. Based on the previous assumption that SIC is selected, the method    assumes no error propagation. This assumption can only be justified    as long as the received SINR constraint in Equation (16) (below) is    selected adequately and is met. The (minimum) received SINR    constraint is equivalent to a (minimum) Rate constraint. This is    discussed further below.-   5. Based on the previous two assumptions, the method assumes that    the ordering of all N_(t) signal elements is based on their    corresponding received Signal-to-Noise & Interference Ratio (SINR),    η_(m), ordered into η₁ _(m,o) , from high to low, where the    interference at the i^(th) iteration is due to the N_(t)−i remaining    information elements. Since the selection of the pre-weighting    elements affects their corresponding received SINR, the ordering of    all N_(t) signal elements based on their corresponding received SINR    is carried out simultaneously with the selection of the    pre-weighting elements.

Constraints “A”:

-   1. In order to minimize (or eliminate) error propagation, a desired    received Signal-to-Noise & Interference Ratio (SINR) for the 1^(st)    element, η₁ _(m,o) , in the ordered received SINR vector {right    arrow over (η)}_(m,o) during the m^(th) iteration should be    constrained to have a lower bound, κ₁ _(m,o) , for 1≦m≦N:

η₁ _(m,o) ≧κ₁ _(m,o)   (16)

Equation (16) is referred to as the (minimum) received SINR constraint.An equivalent performance measure to the ordered received SINR, η₁_(m,o) is the ordered bit Rate, R_(m) _(o) , since both are directlyrelated as follows:

$_{1_{m,o}}\overset{\Delta}{=}{_{m_{o}}{{\log_{2}\left( {1 + \eta_{1_{m,o}}} \right)}.}}$

The importance of such a constraint is to minimize error propagation andto ensure a minimum upload performance for all terminal nodes. Otherminimum performance constraints may also be used.

-   2. Based on the received SINR constraint in Equation (16) for the    1^(st) element, η₁ _(m,o) , in the ordered received SINR vector    {right arrow over (η)}_(m,o) during the m^(th) iteration, or    equivalently for ζ₁ _(m,o) or for α₁ _(m,o) , a pre-weighting    element, γ₁ _(m,o) , is selected such that the following transmit    Power constraint is met:

E{|α ₁ _(1,o) γ₁ _(1,o) |²+ . . . +|α₁ _(Nt,o) γ₁ _(Nt,o) |² }≦P   (17)

where P is a pre-specified upper limit on the total transmitted powerand E{•} denotes statistical averaging. The importance of such aconstraint is to limit the average transmitted power for all terminalnodes. Other power constraints may also be used as described above.

Method “A”:

-   1. If a pre-weighting vector,

$\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix},$

is found which satisfies both the (minimum) received SINR constraint inEquation (16) and the (maximum) transmit Power constraint in Equation(17), then the method optimizes

$\quad\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

such that the sum rate (e.g., in step 911),

₁ _(1,o) + . . . +

₁ _(N,o) =Σ_(m=1) ^(N)

₁ _(m,o) log₂(1+η₁ _(m,o) )   (18)

is increased (or maximized). The sum rate in Equation (18) is differentfrom the one in Equation (3) since n_(m) in Equation (3) is theequivalent received Signal-to-Noise Ratio (SNR) corresponding to them^(th) information element ζ_(m), while η₁ _(m,o) is the SINR valuecorresponding to the 1^(st) element in the ordered received SINR vector{right arrow over (η)}_(m,o) during the m^(th) iteration. This increase(maximization) of the sum rate in Equation (18) can be accomplishedusing an “adjusted” waterfilling strategy. The “regular” waterfillingstrategy optimizes the sum rate with respect to

$\quad\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

under the transmit Power constraint in Equation (17) only. The“adjusted” waterfilling strategy includes the received SINR constraintin Equation (16) together with the transmit Power constraint in Equation(17) when optimizing the sum rate. There are several ways to implementthe “adjusted” waterfilling strategy. For example:

-   -   Way 1. The first way has two steps:        -   a. Find all ordering combinations and corresponding            pre-weighting values which satisfy the received SINR            constraint.        -   b. Select the ordering combination with the largest            corresponding sum rate using, for example, the waterfilling            method subject to the transmit power constraint.    -   Way 2. The second way has two steps:        -   a. Find all ordering combinations and corresponding            pre-weighting values which satisfy the transmit power            constraint (e.g., using step 957).        -   b. Select the ordering combination with the largest            corresponding sum rate using, for example, the waterfilling            method subject to the received SINR constraint.    -   Way 3. The third way has two steps:        -   a. Find all ordering combinations and corresponding            pre-weighting values which satisfy both the received SINR            constraint and the transmit Power constraint (e.g., using            step 978).        -   b. Select the ordering combination with the largest            corresponding sum rate using, for example, two Lagrange            multipliers when obtaining the solution to the sum rate            optimization. The first Lagrange multiplier incorporates the            received SINR constraint, while the second Lagrange            multiplier incorporates the transmit power constraint.    -   Other ways to implement adjusted waterfilling strategies may        also be used. If no ordering combination is found which        satisfies the received SINR constraint and the transmit power        constraint, then use either Method I, II, III or IV below for        Relaxing Received SINR constraint in Method “A.” Other        techniques may also be used to select an ordering combination        and corresponding pre-weighting values when no ordering        combination is found that satisfies the received SINR constraint        and the transmit power constraint.

-   2. If more than one ordering combination for the pre-weighting    vector,

$\quad{\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix},}$

are found which satisfy both the received SINR constraint in Equation(16) and the transmit Power constraint in Equation (17), then the methodselects

$\quad\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

such that the sum rate is optimized (maximized) for each orderingcombination, and selects the ordering combination, which corresponds tothe largest sum rate.

-   3. If a pre-weighting vector,

$\quad{\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix},}$

which satisfies both the received SINR constraint in Equation (16) andthe transmit Power constraint in Equation (17), cannot be found, then,for example, step 923 removes the largest absolute value in

$\quad\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

from

$\quad\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

in Equations (16) and (17) and places it in a set,

_(o). This is repeated until both Equations (16) and (17) are satisfied.The method forces the pre-weighting elements in

_(o) to take a zero value, i.e. their corresponding information elementsare not transmitted.

-   4. Alternatively, the received SINR constraint in Equation (16) is    relaxed (e.g., as described with reference to step 917) by reducing    the set of pre-weighting elements each by a small factor, λ, for    each transmitting device (terminal node). The small factor, λ, is    incremented by a small amount until the transmit Power constraint in    Equation (17) is satisfied (e.g., as described with reference to    step 931). Alternatively, the reduced pre-weighting elements may be    repeatedly reduced by the small factor until the transmit Power    constraint is satisfied.-   5. The method feeds-back (e.g., as described with reference to step    913) the optimized N_(t) pre-weighting elements (whether zero,    non-zero or reduced),

$\quad{\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix},}$

to corresponding U_(t) transmitting devices (terminal nodes). Thefeedback of the pre-weighting values may be communicated using anysuitable method, for example, using PUCCH in an LTE system.

The contributions of the presently disclosed systems and methods includeMethod “A” for the selection of

$\quad\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

selection of and its ordering combination, for an UL MU-MIMO based onMethod “A.”

An Embodiment of the Pre-Weighting Selection in Method “A”:

Since the SIC Method of reception is selected after filtering thereceived signals, then at the i^(th) iteration, the received SINR,corresponding to the 1^(st) ordered element, which is obtained afterfiltering the received signals at all N_(r) Rxs, should comply with theSINR constraint in Equation (16), i.e.

$\begin{matrix}{\eta_{1_{i,o}}\overset{\Delta}{=}{\frac{E\left\{ {{\hat{\alpha}}_{1_{i,o}}^{\prime}}^{2} \right\}}{{E\left\{ {{h_{{Est}_{1_{i,o}}}\begin{bmatrix}\theta_{1_{i,o}} \\\vdots \\\theta_{N_{r_{i,o}}}\end{bmatrix}}}^{2} \right\}} + {E\left\{ {{\sum_{l = 2}^{N_{t} - i + 1}{\hat{\alpha}}_{l_{i,o}}^{\prime}}}^{2} \right\}}} \geq \kappa_{1_{i,o}}}} & (19)\end{matrix}$

where

-   -   η₁ _(i,o) is the ordered received SINR value corresponding to        the 1^(st) ordered signal element for the i ^(th) iteration;    -   κ₁ _(i,o) is the ordered minimum required received SINR value        corresponding to the 1^(st) ordered signal element for the        i^(th) iteration;

${\cdot {\hat{\alpha}}_{1_{i,o}}^{\prime}}\overset{\Delta}{=}{h_{{Est}_{1_{i,o}}}{h_{{Ch}_{i,o}}\begin{bmatrix}{\alpha_{1_{i,o}}\gamma_{1_{i,o}}} \\0 \\\vdots \\0\end{bmatrix}}}$

is the (desired) signal component corresponding to the 1^(st)pre-weighted signal element, α′₁ _(i,o) , in {right arrow over(α)}′_(i,o) after filtering the received signal elements

$\quad\begin{bmatrix}\beta_{1_{i,o}} \\\vdots \\\beta_{N_{r_{i,o}}}\end{bmatrix}$

with a row vector,

h_(Est_(1_(i, o)));

⋅h_(Est_(1_(i, o))) ≡ 1 × N_(r)

is the 1^(st) row vector of the estimation matrix h_(Est) _(i,o) whichis used to estimate the 1^(st) pre-weighted signal element, α′₁ _(i,o) ,in {right arrow over (α)}′_(i,o);

${\cdot \begin{bmatrix}\beta_{1_{i,o}} \\\vdots \\\beta_{N_{r_{i,o}}}\end{bmatrix}} \equiv {N_{r} \times 1}$

is a vector consisting of the received signal elements which remainafter removing the effects of the (i−1) previously detected informationelements;

-   -   h_(Ch) _(i,o) ≡N_(r)×N_(t) is the ordered Channel matrix for the        i^(th) iteration, i.e. the columns of the Channel matrix are        adjusted such that the l^(th) column of h_(Ch) _(i,o)        corresponds to α_(l) _(i,o) :

${\cdot {\hat{\alpha}}_{l_{i,o}}^{\prime}}\overset{\Delta}{=}{h_{{Est}_{1_{i,o}}}{h_{{Ch}_{i,o}}\begin{bmatrix}0 \\\vdots \\0 \\{\alpha_{l_{i,o}}\gamma_{l_{i,o}}} \\0 \\\vdots \\0\end{bmatrix}}}$

is the interference (undesired) component corresponding to l^(th)element, α′_(l) _(i,o) , in {right arrow over (α)}′_(i,o) afterfiltering the received signal elements

$\quad\begin{bmatrix}\beta_{1_{i,o}} \\\vdots \\\beta_{N_{r_{i,o}}}\end{bmatrix}$

with a row vector, h_(Est) _(1i,o) where 2≦l≦N_(t)−i+1;

-   -   Σ_(l=2) ^(N) ^(t) ^(−i+1){circumflex over (α)}′_(l) _(i,o) is        the total interference (undesired) component impinging on the        (desired) pre-weighted signal element {circumflex over (α)}′₁        _(i,o) ; and

$\cdot {h_{{Est}_{1_{i,o}}}\begin{bmatrix}\theta_{1_{i,o}} \\\vdots \\\theta_{N_{r_{i,o}}}\end{bmatrix}}$

is the noise component that results from filtering the noise vector

$\quad\begin{bmatrix}\theta_{1_{i,o}} \\\vdots \\\theta_{N_{r_{i,o}}}\end{bmatrix}$

with the row vector h_(Est) _(1i,o) .

The ordered pre-weighting vector,

${{\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}},$

must be selected to satisfy the received SINR constraint in Equation(19) under the transmit Power constraint in Equation (17). However,since the signal vector,

$\begin{bmatrix}\alpha_{1_{i,o}} \\\vdots \\\alpha_{N_{t_{i,o}}}\end{bmatrix},$

in Equation (19) is unknown to the receiving devices, then twoassumptions are made in Equation (19):

-   -   the elements of the signal vector are independent identically        distributed (iid) with zero mean and variance σ_(α) ² and    -   the elements of the noise are independent identically        distributed (iid) with zero mean and variance σ_(θ) ².

In other words, Equation (19) can re-written as

$\begin{matrix}{\mspace{20mu} {{\eta_{1_{i,o}}\overset{\Delta}{=}{\frac{\alpha_{\alpha}^{2}{\xi_{1_{i,o}}}^{2}}{{\sigma_{\theta}^{2}h_{{Est}_{1_{i,o}}}h_{{Est}_{1_{i,o}}}^{*}} + {\sigma_{\alpha}^{2}{\sum_{l = 2}^{N_{t} - i + 1}{\xi_{l_{i,o}}}^{2}}}} \geq \kappa_{1_{i,o}}}}\mspace{20mu} {where}}} & (20) \\{\mspace{20mu} {{{\xi_{1_{i,o}}}^{2}\overset{\Delta}{=}{h_{{Est}_{1_{i,o}}}{h_{{Ch}_{i,o}}\begin{bmatrix}{\gamma_{1_{i,o}}}^{2} & 0 & \ldots & 0 \\0 & 0 & \ldots & 0 \\\vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & 0\end{bmatrix}}}}\mspace{20mu} {{\xi_{1_{i,o}}}^{2}\overset{\Delta}{=}{h_{{Est}_{1_{i,o}}}{h_{{Ch}_{i,o}}\begin{bmatrix}{\gamma_{1_{i,o}}}^{2} & 0 & \ldots & 0 \\0 & 0 & \ldots & 0 \\\vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & 0\end{bmatrix}}}}}} & \left( {21a} \right) \\{ {{= {{\left\{ {h_{{Est}_{1_{i,o}}}h_{{Ch}_{i,o}}} \right\}_{1}}^{2}{\gamma_{1_{i,o}}}^{2}}}\mspace{20mu} {and}}} & \left( {21b} \right) \\{{\xi_{l_{i,o}}}^{2}\overset{\Delta}{=}{h_{{Est}_{1_{i,o}}}{h_{{Ch}_{i,o}}\begin{bmatrix}0 & \ldots & 0 & 0 & 0 & \ldots & 0 \\\vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\0 & \ldots & 0 & 0 & 0 & \ldots & 0 \\0 & \vdots & \vdots & {\gamma_{l_{i,o}}}^{2} & \vdots & \vdots & \vdots \\0 & \ldots & 0 & 0 & 0 & \ldots & 0 \\\vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\0 & \ldots & 0 & 0 & 0 & \ldots & 0\end{bmatrix}}h_{{Ch}_{i,o}}^{*}h_{{Est}_{1_{i,o}}}^{*}}} & \left( {22a} \right) \\{\mspace{65mu} {\overset{\Delta}{=}{{\left\{ {h_{{Est}_{1_{i,o}}}h_{{Ch}_{i,o}}} \right\}_{l}}^{2}{\gamma_{l_{i,o}}}^{2}}}} & \left( {22b} \right)\end{matrix}$

where

{h_(Est_(1_(i, o)))h_(Ch_(i, o))}_(l)

is the l^(th) element in h_(Est) _(1i,o) h_(Ch) _(i,o) . If theestimation filter,

h_(Est_(1_(i, o))),

is normalized, i.e.

h_(Est_(1_(i, o)))h_(Est_(1_(i, o)))^(*) = 1,

then

$\begin{matrix}{\eta_{1_{i,o}}\overset{\Delta}{=}{\frac{\sigma_{\alpha}^{2}{\xi_{1_{i,o}}}^{2}}{\sigma_{\theta}^{2} + {\sigma_{\alpha}^{2}{\sum_{l = 2}^{N_{t} - i + 1}{\xi_{l_{i,o}}}^{2}}}} \geq \kappa_{1_{i,o}}}} & \left( {23a} \right) \\{= {\frac{{\left\{ {h_{{Est}_{1_{i,o}}}h_{{Ch}_{i,o}}} \right\}_{1}}^{2}{\gamma_{1_{i,o}}}^{2}}{\frac{1}{v} + {\sum_{l = 2}^{N_{t} - i + 1}{{\left\{ {h_{{Est}_{1_{i,o}}}h_{{Ch}_{i,o}}} \right\}_{l}}^{2}{\gamma_{l_{i,o}}}^{2}}}} \geq \kappa_{1_{i,o}}}} & \left( {23b} \right)\end{matrix}$

In this case, the transmit Power constraint can be re-written as

$\begin{matrix}{{E\left\{ {{{\alpha_{1_{1,o}}\gamma_{1_{1,o}}}}^{2} + \ldots + {{\alpha_{1_{N_{t},o}}\gamma_{1_{N_{t},o}}}}^{2}} \right\}} = {{\sigma_{\alpha}^{2}\left\{ {{\gamma_{1_{1,o}}}^{2} + \ldots + {\gamma_{1_{N_{t},o}}}^{2}} \right\}} \leq P}} & (24)\end{matrix}$

where σ_(α) ² is known from the transmit power and modulation format.

If the pre-weighting vector, {right arrow over (γ)}_(o), is found tosatisfy both the received SINR constraint in Equation (23) and thetransmit Power constraint in Equation (24), then the next step is tooptimize the sum rate in Equation (18). This can be accomplished usingthe “adjusted” waterfilling strategy as explained above using Way 1, Way2 or Way 3.

A Solution of Equation (23) in Method “A”:

When i=N_(t), Equation (23) reduces to

${\frac{\sigma_{\alpha}^{2}{\xi_{1_{N_{t},o}}}^{2}}{\sigma_{\theta}^{2}} \geq \kappa_{1_{N_{t},o}}},$

or equivalently

$\begin{matrix}{{\xi_{1_{N_{t},o}}}^{2} \geq \frac{\kappa_{1_{N_{t},o}}}{v}} & \left( {25a} \right)\end{matrix}$

where

$v\overset{\Delta}{=}{\sigma_{\theta}^{2}\text{/}\sigma_{a}^{2}}$

is the normalized SNR corresponding to the transmitted signal elements.

When i=N_(t)−1, Equation (23) reduces to

${\frac{\sigma_{\alpha}^{2}{\xi_{1_{{N_{t} - 1},o}}}^{2}}{\sigma_{\theta}^{2} + {\sigma_{\alpha}^{2}{\xi_{2_{{N_{t} - 1},o}}}^{2}}} \geq \kappa_{1_{{N_{t} - 1},o}}},$

or equivalently

$\begin{matrix}{{\xi_{1_{{N_{t} - 1},o}}}^{2} \geq {\kappa_{1_{{N_{t} - 1},o}}\left( {\frac{1}{v} + {\xi_{2_{{N_{t} - 1},o}}}^{2}} \right)}} & \left( {25b} \right)\end{matrix}$

When i=N_(t)−2, Equation (23) reduces to

${\frac{\sigma_{\alpha}^{2}{\xi_{1_{{N_{t} - 1},o}}}^{2}}{\sigma_{\theta}^{2} + {\sigma_{\alpha}^{2}{\xi_{2_{{N_{t} - 1},o}}}^{2}} + {\sigma_{\alpha}^{2}{\xi_{3_{{N_{t} - 2},o}}}^{2}}} \geq \kappa_{1_{{N_{t} - 2},o}}},$

or equivalently

$\begin{matrix}{{\xi_{1_{{N_{t} - 2},o}}}^{2} \geq {\kappa_{1_{{N_{t} - 2},o}}\left( {\frac{1}{v} + {\xi_{2_{{N_{t} - 2},o}}}^{2} + {\xi_{3_{{N_{t} - 2},o}}}^{2}} \right)}} & \left( {25c} \right)\end{matrix}$

In general, at the i^(th) iteration, we have

$\begin{matrix}{{\xi_{1_{i,o}}}^{2} \geq {\kappa_{1_{i,o}}\left( {\frac{1}{v} + {\xi_{2_{i,o}}}^{2} + {\xi_{3_{i,o}}}^{2} + \ldots + {\xi_{N_{t} - i + 1_{i,o}}}^{2}} \right)}} & \left( {25d} \right)\end{matrix}$

for 1≦i≦N_(t). From Equation (25a), |γ₁ _(Nt,o) |² can be derived as

$\begin{matrix}{{\gamma_{1_{N_{t},o}}}^{2} \geq \frac{\kappa_{1_{N_{t},o}}}{v{\left\{ {h_{{Est}_{1_{N_{t},o}}}h_{{Ch}_{N_{t},o}}} \right\}_{1}}^{2}}} & \left( {26a} \right)\end{matrix}$

From Equation (25b), |γ₁ _(Nt−1,o) |² can be derived as

$\begin{matrix}{{\gamma_{1_{{N_{t} - 1},o}}}^{2} \geq {\frac{\kappa_{1_{{N_{t} - 1},o}}}{{\left\{ {h_{{Est}_{1_{{N_{t} - 1},o}}}h_{{Ch}_{{N_{t} - 1},o}}} \right\}_{1}}^{2}}\left( {\frac{1}{v} + {\xi_{2_{{N_{t} - 1},o}}}^{2}} \right)}} & \left( {26b} \right)\end{matrix}$

In general, from Equation (25d), |₁ _(i,o) |² can be derived as

$\begin{matrix}{{\gamma_{1_{i,o}}}^{2} \geq {\frac{\kappa_{1_{i,o}}}{{\left\{ {h_{{Est}_{1_{i,o}}}h_{{Ch}_{i,o}}} \right\}_{1}}^{2}}\left( {\frac{1}{v} + {\xi_{2_{i,o}}}^{2} + {\xi_{3_{i,o}}}^{2} + \ldots + {\xi_{N_{t} - i + 1_{i,o}}}^{2}} \right)}} & \left( {26d} \right)\end{matrix}$

After deriving the pre-weighting vector,

$\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix},$

the transmit Power constraint in Equation (24) is tested as follows

$\begin{matrix}{{{\gamma_{1_{1,o}}}^{2} + \ldots + {\gamma_{1_{N_{t},o}}}^{2}} \leq \frac{P}{\sigma_{\alpha}^{2}}} & (27)\end{matrix}$

A Selection of the Ordering of

$\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{t}}\end{bmatrix}\mspace{14mu} {{into}\mspace{14mu}\begin{bmatrix}\eta_{1_{1,o}} \\\vdots \\\eta_{1_{N_{t},o}}\end{bmatrix}}$

in Method “A”:

Based on Equations (26) and the transmit Power constraint in Equation(27), the ordering of

$\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{t}}\end{bmatrix}\mspace{14mu} {{into}\mspace{14mu}\begin{bmatrix}\eta_{1_{1,o}} \\\vdots \\\eta_{1_{N_{t},o}}\end{bmatrix}}$

is based on selecting

$\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}\quad$

in such a way as to maximize the sum rate under the followingconstraints:

$\begin{matrix}{{\sum\limits_{i = 1}^{N_{t}}\left\{ {\frac{\kappa_{1_{i,o}}}{{\left\{ {h_{{Est}_{1_{i,o}}}h_{{Ch}_{i,o}}} \right\}_{1}}^{2}}\left( {\frac{1}{v} + {\sum\limits_{l = 2}^{N_{t} - i + 1}{\xi_{l_{i,o}}}^{2}}} \right)} \right\}} \leq {\sum\limits_{i = 1}^{N_{t}}{\gamma_{1_{i,o}}}^{2}} \leq \frac{P}{\sigma_{\alpha}^{2}}} & \left( {28a} \right)\end{matrix}$

One possible way for ordering

$\begin{bmatrix}\eta_{1_{1,o}} \\\vdots \\\eta_{1_{N_{t},o}}\end{bmatrix}\quad$

and for selecting

$\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}\quad$

which satisfies the constraints in Equation (28a), is to exhaustivelysearch for all possible ordering combinations of

$\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}\quad$

until at least one ordering combination satisfies Equation (28b)

$\begin{matrix}\begin{matrix}{{\sum\limits_{i = 1}^{N_{t}}\left\{ {\frac{\kappa_{1_{i,o}}}{{\left\{ {h_{{Est}_{1_{i,o}}}h_{{Ch}_{i,o}}} \right\}_{1}}^{2}}\left( {\frac{1}{v} + {\sum\limits_{k = 2}^{N_{t} - i + 1}{\xi_{k_{i,o}}}^{2}}} \right)} \right\}} \leq \frac{P}{\sigma_{\alpha}^{2}}} & \;\end{matrix} & \left( {28b} \right)\end{matrix}$

If one ordering combination satisfies Equation (28b), then the next stepis to select

$\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}\quad$

in Equation (28a) in such a way that the sum rate is maximized using an“adjusted” waterfilling as explained above. Adjusted waterfilling may beused to find a solution with better capacity after a satisfactorysolution is found. If more than one ordering combinations satisfyEquation (28b), then the next step is to select the ordering, whichcorresponds to the largest minimum sum rate.

Another possible way for ordering

$\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}\quad$

(and for selecting

$\begin{bmatrix}\eta_{1_{1,o}} \\\vdots \\\eta_{1_{N_{t},o}}\end{bmatrix}\quad$

) which satisfies the constraints in Equation (28b), is to sort

$\begin{bmatrix}\frac{\kappa_{1}}{\left\{ {h_{{Est}_{1}}h_{Ch}} \right\}_{1}} \\\vdots \\\frac{\kappa_{N_{t}}}{\left\{ {h_{{Est}_{N_{t}}}h_{Ch}} \right\}_{1}}\end{bmatrix}\quad$

from high to low with the largest value assigned to i=N_(t), and thenext largest value assigned to i=N_(t)−1, etc., where

-   -   h_(Est) _(m) ≡1×N_(r) is the m^(th) row vector of the estimation        matrix h_(Est) which is used to estimate the m^(th) pre-weighted        signal element, α′_(m), in {right arrow over (α)}′; and    -   {h_(Est) _(m) h_(Ch)}₁≡1×1 is the scalar value that corresponds        to the first element in the row vector h_(Est) _(m) h_(Ch).

Once again, if one ordering combination, that is based on sorting

$\begin{bmatrix}\frac{\kappa_{1}}{\left\{ {h_{{Est}_{1}}h_{Ch}} \right\}_{1}} \\\vdots \\\frac{\kappa_{N_{t}}}{\left\{ {h_{{Est}_{N_{t}}}h_{Ch}} \right\}_{1}}\end{bmatrix}\quad$

from high to low, satisfies Equation (28b), then the next step is toselect

$\quad\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

in Equation (28a) in such a way that the sum rate is maximized using an“adjusted” waterfilling. On the other hand, if more than one orderingcombinations, that are based on sorting

$\quad\begin{bmatrix}\frac{\kappa_{1}}{\left\{ {h_{{Est}_{1}}h_{Ch}} \right\}_{1}} \\\vdots \\\frac{\kappa_{N_{t}}}{\left\{ {h_{{Est}_{N_{t}}}h_{Ch}} \right\}_{1}}\end{bmatrix}$

from high to low, satisfy Equation (28b), then the next step is toselect the ordering, which corresponds to the largest minimum sum rate.

Method I for Relaxing Received SINR Constraint in Method “A”:

Alternatively, if Equation (28b) cannot be satisfied for any ordering of

$\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{t}}\end{bmatrix},$

several remedies exist. For example, the maximum value, |γ₁ _(S,o) |²,which is obtained as

${{\gamma_{1_{,o}}}^{2}\overset{\Delta}{=}{\max \left\{ {{\gamma_{1_{1,o}}}^{2},{\gamma_{1_{2,o}}}^{2},\ldots \mspace{14mu},{\gamma_{1_{{N_{t} - 1},o}}}^{2}} \right\}}},$

is removed (e.g., as described with reference to step 923) from the lefthand side of Equation (28b) and placed in a set,

, and its corresponding indices,

is placed in another set,

_(o).

This is repeated until Equation (28b) is satisfied. The formed set ofsquared values in

, is then replaced by a zero value, i.e. the signal elements thatcorrespond to

are not transmitted.

Method II for Relaxing Received SINR Constraint in Method “A”:

Another remedy for the case when Equation (28b) cannot be satisfied forany ordering of

$\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{t}}\end{bmatrix},$

is to reduce (e.g., as described with reference to step 927) the lefthand side of Equation (28b) by a factor which would make it equal to theright hand side of equation (28b), i.e. we need to find a factor, λ,such that

${\lambda {\sum\limits_{i = 1}^{N_{t}}\left\{ {\frac{\kappa_{1_{i,o}}}{{\left\{ {h_{{Est}_{1_{i,o}}}h_{{Ch}_{i,o}}} \right\}_{1}}^{2}}\left( {\frac{1}{v} + {\sum\limits_{k = 2}^{N_{t} - i + 1}{\xi_{k_{i,o}}}^{2}}} \right)} \right\}}} = \frac{P}{\sigma_{\alpha}^{2}}$

In other words, instead of accommodating only a few of the terminalnodes as in the previous strategy, this strategy attempts to accommodateall terminal nodes in a fair fashion. Other methods for relaxing thereceived SINR constraint in Method “A” may also be used.

Note: When choosing between the two Methods for relaxing the receivedSINR constraint, one can rely on the variance of {|γ₁ _(1,o) |², |γ₁_(2,o) |², . . . , |γ₁ _(Nt−1,o) |²}. If the variance is larger than apre-specified threshold (e.g., as described with reference to step 919),then Method I is selected, otherwise, Method II is selected. There aremany ways for selecting the pre-specified threshold. For example, thepre-specified threshold can be selected as a function of the variance of{|γ₁ _(1,o) |², |γ₁ _(2,o) |², . . . , |γ₁ _(Nt−1,o) |²}. An example ofsuch a function is a normalized mean or a normalized median. Othercriteria of choosing between Method I and Method II may also be used. Inalternative embodiments, only Method I or Method II may be used forrelaxing the received SINR constraint.

Example for Method “A”:

An example of a 2×2 UL MU-MIMO is used to describe Method “A.” In thisexample, the following 2×2 matrix is selected to describe a channel:

$h_{ch} = \begin{bmatrix}{0.22 - {0.52j}} & {0.94 - {0.63j}} \\{0.20 + {0.20j}} & {0.48 - {0.29j}}\end{bmatrix}$

which is assumed to consist of two Txs: Tx₁ and Tx₂ and two Rxs: Rx₁ andRx₂. The signal transmitted by Tx₂ and received at Rx₁ has 6 dB morepower than the signal transmitted by Tx₂ and received at Rx₂ or thesignal transmitted by Tx₁ and received at Rx₁. The signal transmitted byTx₂ and received at Rx₁ has 12 dB more power than the signal transmittedby Tx₁ and received at Rx₂. There are two possible orderingcombinations:

-   -   Order 1: Perform SIC on Tx₁ first, then on Tx₂; or    -   Order 2: Perform SIC on Tx₂ first, then on Tx₁,

If κ₁=κ₂=10, then Order 1 requires a ratio

${\frac{\gamma_{1}}{\gamma_{2}} = 8.71},$

while Order 2 requires only a ratio of

$\frac{\gamma_{1}}{\gamma_{2}} = {1.8.}$

When σ_(α) ²=1 and σ_(θ) ²=0.1, then Order 1 offers a received SINR of12.2 dB>κ₁ & κ₂ for both received signals, while Order 2 offers areceived SINR of 14.1 dB>κ₁ & κ₂ also for both received signals. Thistranslates to a sum rate for Order 1 which is equal to 8.2 bps/Hz, and asum rate for Order 2 equal to 9.5 bps/Hz. If Order 2 satisfies thereceived SINR constraint, then, γ₁ and γ₂ can be re-adjusted to increase(maximize) the sum rate to 9.62 bps/Hz by allowing

$\frac{\gamma_{1}}{\gamma_{2}} = 1.$

In conclusion, by selecting Order 2, one can offer a larger sum ratethan the one offered by Order 1.

Another Embodiment for Pre-Weighting Selection in Method “A”:

For the special case of UL MU-MIMO where N_(r)=1, i.e. all receivingdevices have a total of only one antenna, UL MU-MIMO reduces to ULMU-MISO. In this case, the set of ordered received SINR equations iswritten as:

$\begin{matrix}{\eta_{1_{1,o}}\overset{\Delta}{=}{\frac{E\left\{ {{h_{1_{1,o}}h_{{Ch}_{1_{1,o}}}\gamma_{1_{1,o}}\alpha_{1_{1,o}}}}^{2} \right\}}{{E\left\{ {{h_{1_{{1,o}\;}}\theta_{1_{1,o}}}}^{2} \right\}} + {E\left\{ {{\sum_{l = 2}^{N_{t}}{h_{1_{1,o}}h_{{Ch}_{l_{1,o}}}\gamma_{l_{1,o}}\alpha_{l_{1,o}}}}}^{2} \right\}}} \geq \kappa_{1_{1,o}}}} & \left( {29a} \right) \\{\eta_{1_{2,o}}\overset{\Delta}{=}{\frac{E\left\{ {{h_{1_{2,o}}h_{{Ch}_{1_{2,o}}}\gamma_{1_{2,o}}\alpha_{1_{2,o}}}}^{2} \right\}}{{E\left\{ {{h_{1_{{2,o}\;}}\theta_{1_{1,o}}}}^{2} \right\}} + {E\left\{ {{\sum_{l = 3}^{N_{t}}{h_{1_{2,o}}h_{{Ch}_{l_{{o\; 2},}}}\gamma_{l_{2,o}}\alpha_{l_{2,o}}}}}^{2} \right\}}} \geq \kappa_{1_{2,o}}}} & \left( {29b} \right) \\{\mspace{20mu} \vdots} & \vdots \\{\mspace{20mu} {\eta_{1_{N_{t},o}}\overset{\Delta}{=}{\frac{E\left\{ {{h_{1_{N_{t}},o}h_{{Ch}_{1_{N_{t},o}}}\gamma_{1_{N_{t},o}}\alpha_{1_{N_{t},o}}}}^{2} \right\}}{E\left\{ {{h_{1_{N_{t},o}}\theta_{1_{1,o}}}}^{2} \right\}} \geq \kappa_{1_{N_{t}},o}}}} & \left( {29c} \right)\end{matrix}$

where

$\bullet \begin{bmatrix}\eta_{1_{1,o}} \\\vdots \\\eta_{1_{N_{t},o}}\end{bmatrix}$

is the ordered received SINR vector;

$\bullet \begin{bmatrix}\kappa_{1_{1,o}} \\\vdots \\\kappa_{1_{N_{t}},o}\end{bmatrix}$

is the ordered minimum required received SINR;

$\bullet \begin{bmatrix}h_{1_{1,o}} \\\vdots \\h_{1_{N_{t},o}}\end{bmatrix}$

is the ordered filter element;

$\bullet \begin{bmatrix}h_{{Ch}_{1_{1,o}}} \\\vdots \\h_{{Ch}_{1_{N_{t},o}}}\end{bmatrix}$

is the ordered Channel vector, h_(Ch);

$\quad\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

is the ordered pre-weighting vector, {right arrow over (γ)}_(o); and

$\quad\begin{bmatrix}\alpha_{1_{1,o}} \\\vdots \\\alpha_{1_{N_{t},o}}\end{bmatrix}$

is the ordered signal vector, {right arrow over (α)}_(o).

The pre-weighting vector, {right arrow over (γ)}₀, must be selected tosatisfy the received SINR constraints in Equations (29a), . . . , (29c),under the transmit Power constraint in Equation (26). However, since thesignal vector,

$\quad{\begin{bmatrix}\alpha_{1_{1,o}} \\\vdots \\\alpha_{1_{N_{t},o}}\end{bmatrix},}$

is unknown to the receiving devices, then an assumption in Equations(29a), . . . , (29c) is made, which is to assume that the elements ofthe signal vector are independent identically distributed (iid) withzero mean and variance σ_(α) ². In other words, Equations (29a), . . . ,(29c) can re-written as

$\begin{matrix}{\eta_{1_{1,o}}\overset{\Delta}{=}{\frac{\sigma_{\alpha}^{2}{{h_{{Ch}_{1_{1,o}}}\gamma_{1_{1,o}}}}^{2}}{\sigma_{\theta}^{2} + {\sigma_{\alpha}^{2}{\sum_{l = 2}^{N_{t}}{{h_{{Ch}_{l_{1,o}}}\gamma_{l_{1,o}}}}^{2}}}} \geq \kappa_{1_{1,o}}}} & \left( {30a} \right) \\{\eta_{1_{2,o}}\overset{\Delta}{=}{\frac{\sigma_{\alpha}^{2}{{h_{{Ch}_{1_{2,o}}}\gamma_{1_{2,o}}}}^{2}}{\sigma_{\theta}^{2} + {\sigma_{\alpha}^{2}{\sum_{l = 3}^{N_{t}}{{h_{{Ch}_{l_{2,o}}}\gamma_{l_{2,o}}}}^{2}}}} \geq \kappa_{1_{2,o}}}} & \left( {30b} \right) \\\vdots & \vdots \\{\eta_{1_{N_{t},o}}\overset{\Delta}{=}{\frac{\sigma_{\alpha}^{2}{{h_{{Ch}_{1_{N_{t},o}}}\gamma_{1_{N_{t},o}}}}^{2}}{\sigma_{\theta}^{2}} \geq \kappa_{1_{N_{t},o}}}} & \left( {30c} \right)\end{matrix}$

Or equivalently, {right arrow over (γ)}_(o)

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

can be derived starting with Equation (30c) as follows

$\begin{matrix}{{{h_{{Ch}_{1_{N_{t},o}}}\gamma_{1_{N_{t},o}}}}^{2} \geq \frac{\kappa_{1_{N_{t},o}}}{v}} & \left( {31a} \right) \\{{{h_{{Ch}_{1_{{N_{t} - 1},o}}}\gamma_{1_{{N_{t} - 1},o}}}}^{2} \geq {\frac{\kappa_{1_{{N_{t} - 1},o}}}{v}\left( {1 + \kappa_{1_{N_{t},o}}} \right)}} & \left( {31b} \right) \\{{{h_{{Ch}_{1_{{N_{t} - 2},o}}}\gamma_{1_{{N_{t} - 2},o}}}}^{2} \geq {\frac{\kappa_{1_{{N_{t} - 2},o}}}{v}\left( {1 + \kappa_{1_{N_{t - 1},o}}} \right)\left( {1 + \kappa_{1_{N_{t},o}}} \right)}} & \left( {31c} \right) \\\vdots & \vdots \\{{{h_{{Ch}_{1_{1,o}}}\gamma_{1_{1,o}}}}^{2} \geq {\frac{\kappa_{1_{1,o}}}{v}\left( {1 + \kappa_{1_{2,o}}} \right)\left( {1 + \kappa_{1_{3,0}}} \right)\mspace{14mu} \ldots \mspace{14mu} \left( {1 + \kappa_{1_{N_{t},o}}} \right)}} & \left( {31c} \right)\end{matrix}$

In general, we have N_(t) equations to be used to derive

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

as follows

$\begin{matrix}{{\gamma_{1_{i,o}}}^{2} \geq {\frac{\kappa_{1_{i,o}}}{v{h_{{Ch}_{1_{i,o}}}}^{2}}{\prod_{k = {i + 1}}^{N_{t}}{\left( {1 + \kappa_{1_{k,o}}} \right)\mspace{14mu} 1}}} \leq  \leq N_{t}} & (32)\end{matrix}$

where

$v\overset{\Delta}{=}\frac{\sigma_{\alpha}^{2}}{\sigma_{\theta}^{2}}$

is the normalized SNR for the signal elements.

Another Selection of the Ordering of

$\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{t}}\end{bmatrix}\mspace{14mu} {{into}\mspace{14mu}\begin{bmatrix}\eta_{1_{1,o}} \\\vdots \\\eta_{1_{N_{t},o}}\end{bmatrix}}$

in Method “A”:

Based on Equation (32) and the transmit Power constraint in Equation(26), a selection of the ordering of

$\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{t}}\end{bmatrix}\mspace{14mu} {{into}\mspace{14mu}\begin{bmatrix}\eta_{1_{1,o}} \\\vdots \\\eta_{1_{N_{t},o}}\end{bmatrix}}$

in Equation (30) is based on selecting |γ₁ _(i,o) |² in such a way as tomaximize the sum rate under the constraints:

$\begin{matrix}{{\sum_{i = 1}^{N_{t}}\left\{ {\frac{\kappa_{1_{i,o}}}{v{h_{{Ch}_{1_{i,o}}}}^{2}}{\prod_{k = {i + 1}}^{N_{t}}\left( {1 + \kappa_{1_{k,o}}} \right)}} \right\}} \leq {\sum_{i = 1}^{N_{t}}{\gamma_{1_{i,o}}}^{2}} \leq \frac{P}{\sigma_{\alpha}^{2}}} & \left( {33a} \right)\end{matrix}$

One possible way for ordering

$\quad\; \begin{bmatrix}\eta_{1_{1,o}} \\\vdots \\\eta_{1_{N_{t},o}}\end{bmatrix}$

and for selecting

$\quad\; \begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

which satisfies the constraints in Equation (33a), is to exhaustivelysearch for all possible ordering combinations of

$\quad\; \begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

until at least one ordering combination satisfies Equation (33b)

$\begin{matrix}{{\sum_{i = 1}^{N_{t}}\left\{ {\frac{\kappa_{1_{i,o}}}{v{h_{{Ch}_{1_{i,o}}}}^{2}}{\Pi_{k = {i + 1}}^{N_{t}}\left( {1 + \kappa_{1_{k,o}}} \right)}} \right\}} \leq \frac{P}{\sigma_{\alpha}^{2}}} & \left( {33b} \right)\end{matrix}$

If one ordering combination satisfies Equation (33b), then the next stepis to select

$\quad\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

in Equation (33a) in such a way that the sum rate is maximized using an“adjusted” waterfilling strategy. If more than one ordering combinationssatisfy Equation (33b), then the next step is to select the ordering,which corresponds to the largest minimum sum rate.

Another possible way for ordering

$\quad\begin{bmatrix}\eta_{1_{1,o}} \\\vdots \\\eta_{1_{N_{t},o}}\end{bmatrix}$

(and for selecting

$\quad\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

) which satisfies the constraints in Equation (33b), is to sort

$\quad\begin{bmatrix}\frac{\kappa_{1}}{{h_{{Ch}_{1}}}^{2}} \\\vdots \\\frac{\kappa_{N_{t}}}{{h_{{Ch}_{N_{t}}}}^{2}}\end{bmatrix}$

from high to low with the largest value assigned to i=N_(t), and thenext largest value assigned to i=N_(t)−1, etc. Once again, if oneordering combination satisfies Equation (33b), then the next step is toselect

$\quad\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

in Equation (33a) in such a way that the sum rate is maximized using an“adjusted” waterfilling strategy. On the other hand, if more than oneordering combinations satisfy Equation (33b), then the next step is toselect the ordering, which corresponds to the largest minimum sum rate.

Method III for Relaxing Received SINR Constraint in Method “A”:

Alternatively, if Equation (33b) cannot be satisfied for any ordering of

$\quad{\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{t}}\end{bmatrix},}$

several remedies exist (e.g., using process 917 of FIG. 9F). Forexample, the maximum value, |γ₁ _(S,o) |², which is obtained as

${{\gamma_{1_{,o}}}^{2}\overset{\Delta}{=}{\max \left\{ {{\gamma_{1_{1,o}}}^{2},{\gamma_{1_{2,o}}}^{2},\ldots \mspace{14mu},{\gamma_{1_{{N_{t} - 1},o}}}^{2}} \right\}}},$

is removed (e.g., as described with reference to step 923) from the lefthand side of Equation (33b) and is placed in a set,

, and its corresponding indices,

is placed in another set,

_(o). This is repeated until Equation (33b) is satisfied. The formed setof squared values in

, is then replaced by a zero value, i.e. the corresponding informationelements that correspond to

are not transmitted.

Method IV for Relaxing Received SINR Constraint in Method “A”:

Another remedy for the case when Equation (33b) cannot be satisfied forany ordering of

$\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{t}}\end{bmatrix},$

is to reduce (e.g., as described with reference to step 927) the lefthand side of Equation (33b) by a factor which would make it equal to theright hand side of equation (33b), i.e. we need to find a factor, λ,such that

${\lambda {\sum_{i = 1}^{N_{t}}\left\{ {\frac{\kappa_{1_{i,o}}}{v{h_{{Ch}_{1_{i,o}}}}^{2}}{\sum\limits_{k = {i + 1}}^{N_{t}}\left( {1 + \kappa_{1_{k,o}}} \right)}} \right\}}} = \frac{P}{\sigma_{\alpha}^{2}}$

In other words, instead of accommodating only a few of the terminalnodes as in the previous strategy, this strategy attempts to accommodateall terminal nodes in a fair fashion.

Note: When choosing between the two Methods for relaxing the SINRconstraint, one can rely on the variance of {|γ₁ _(1,o) |², |γ₁ _(2,o)|², . . . , |γ₁ _(Nt−1,o) |²}. If the variance is above a pre-specifiedthreshold (e.g., as described with reference to step 919), then MethodIII is selected, otherwise, Method IV is selected.

DL MU-MIMO Network:

We now consider a Downlink Multi-User MIMO (DL MU-MIMO) network. Sincethe network is an MU network, we assume that a number, U_(t), oftransmitting devices (access nodes) are assigned to communicatesimultaneously with a number, U_(r), of receiving devices (terminalnodes). Since the network is a MIMO network, we assume that the v^(th)transmitting device (access node) contains a number, N_(t) ^(v), oftransmit antennas and that the w^(th) receiving device (terminal node)contains a number, N_(r) ^(w), of receive antennas. In other words, theU_(t) transmitting devices (access nodes) transmit N_(t) pre-weightedsignal elements, {right arrow over (α)}′, at a time (i.e. during oneepoch), which represent N information elements, {right arrow over (ζ)},across a communications channel over a total of N_(t) transmit antennas(N_(t) Txs), and the U_(r) receiving devices (access nodes) receive theN_(t) transmitted pre-weighted signal elements over a total of N_(r)receive antennas (N_(r) Rxs), where

$N_{t}\overset{\Delta}{=}{{N_{t}^{1} + \ldots + {N_{t}^{U_{t}}\mspace{14mu} {and}\mspace{14mu} N_{r}}}\overset{\Delta}{=}{N_{r}^{1} + \ldots + {N_{r}^{U_{r}}.}}}$

The v^(th) transmitting device (access node) intends to transmit aninformation element, ζ_(k,m) ^(w,v) using its m^(th) Tx to the k^(th) Rxof the w^(th) receiving device (terminal node) over a communicationschannel defined by a channel element, h_(Ch) _(k,m) ^(w,v). The sum rateof the system is therefore

Σ_(w=1) ^(U) ^(r) Σ_(v=1) ^(U) ^(t) Σ_(k=1) ^(N) ^(r) ^(w) Σ_(m=1) ^(N)^(t) ^(v)

_(k,m) ^(w,v)=Σ_(w=1) ^(U) ^(r) Σ_(v=1) ^(U) ^(t) Σ_(k=1) ^(N) ^(r) ^(w)Σ_(m=1) ^(N) ^(t) ^(v)

_(k,m) ^(w,v) log₂(1+η_(k,m) ^(w,v))   (34)

where

-   -   _(k,m) ^(w,v) is the rate corresponding to the transmission by        the v^(th) transmitting device (access node) of the information        element, ζ_(k,m) ^(w,v) using its m^(th) Tx to the k^(th) Rx of        the w^(th) receiving device (terminal node);    -   _(k,m) ^(w,v) is the bandwidth corresponding to the transmission        by the v^(th) transmitting device (access node) of the        information elements, ζ_(k,m) ^(w,v) using its m^(th) Tx to the        k^(th) Rx of the w^(th) receiving device (terminal node); and    -   η_(k,m) ^(w,v) is the received SINR for the information        elements, ζ_(k,m) ^(w,v) at the k^(th) Rx of the w^(th)        receiving device (terminal node).

Equation (34) can be used to represent the sum rate corresponding to anytype of downlink transmissions including multicasting, unicasting andbroadcasting transmissions. Equivalently, the U_(t) transmitting devices(access nodes) transmit U_(t) signal vectors, {right arrow over (α)}″¹,. . . , {right arrow over (α)}″^(U) ^(t) , at a time, i.e. a signalvector per transmitting device, to U_(r) receiving devices. The w^(th)receiving device (terminal node) receives a signal vector, {right arrowover (β)}^(w), consisting of N_(r) ^(w) elements, over N_(r) ^(w) Rxs,i.e. one received signal element per Rx. The communications channelbetween all N_(t) TXs (access node antennas) and the N_(r) ^(w) Rxs(w^(th) terminal node antennas) is assumed to be flat fading lineartime-invariant (LTI) with additive white Gaussian Noise (AWGN), i.e. thechannel can be characterized using the sub-matrices, h_(Ch) ^(w,1), . .. , h_(Ch) ^(w,U) ^(r) and the noise can be characterized using a noisevector, {right arrow over (θ)}^(w). The relationship between

${\overset{\rightarrow}{\beta}}^{w}\mspace{14mu} {{and}\mspace{14mu}\begin{bmatrix}{\overset{\rightarrow}{\alpha}}^{''\; 1} \\\vdots \\{\overset{\rightarrow}{\alpha}}^{''\; U_{t}}\end{bmatrix}}$

during one epoch is therefore assumed to be

$\begin{matrix}{{\overset{\rightarrow}{\beta}}^{w} = {{\sum_{v = 1}^{U_{r}}{h_{Ch}^{w,v}\alpha^{''\; v}}} + {\overset{\rightarrow}{\theta}}^{w}}} & \left( {35a} \right) \\{\mspace{31mu} {= {{\begin{bmatrix}h_{Ch}^{w,1} & \ldots & h_{Ch}^{w,U_{r}}\end{bmatrix}\begin{bmatrix}{\overset{\rightarrow}{\alpha}}^{{\prime\prime}\; 1} \\\vdots \\{\overset{\rightarrow}{\alpha}}^{{\prime\prime}\; U_{t}}\end{bmatrix}} + {\overset{\rightarrow}{\theta}}^{w}}}} & \left( {35b} \right) \\{\mspace{31mu} {= {{h_{Ch}^{w}{\overset{\rightarrow}{\alpha}}^{''}} + {\overset{\rightarrow}{\theta}}^{w}}}} & \left( {35c} \right)\end{matrix}$

where

-   -   h_(Ch) ^(w,v)≡N_(r) ^(w)×N_(t) ^(v) is the Channel sub-matrix        between the v^(th) transmitting device (access node) and the        w^(th) receiving device (terminal node), which is defined by its        k^(th) row and m^(th) column element, h_(Ch) _(k,m) ^(w,v);    -   h_(Ch) _(k,m) ^(w,v) represents the complex attenuation of the        flat fading linear time-invariant wireless channel connecting        the m^(th) Tx (access node antenna) of the v^(th) transmitting        device (access node) to the k^(th) Rx (terminal node receive        antenna) of the w^(th) receiving device (terminal node) for        1≦m≦N_(t) ^(v), 1≦k≦N_(r) ^(w), 1≦v≦U_(t), and 1≦w≦U_(r);    -   {right arrow over (α)}″≡N_(t)×1 is the pre-coded and        pre-weighted signal vector for the N_(t) transmitting devices        (access nodes), which is defined by its m^(th) element, α″_(m);    -   α″_(m) is the m^(th) element of {right arrow over (α)}″ which is        defined as

${{\overset{\rightarrow}{\alpha}}^{''}\overset{\Delta}{=}{\chi \left\{ {\overset{\rightarrow}{\alpha}}^{\prime} \right\}}};$

-   -   {right arrow over (α)}′≡N_(t)×1 is the pre-weighted signal        vector which is defined by its m^(th) pre-weighted signal        element, α′_(m);    -   α′_(m) is the m^(th) element of {right arrow over (α)}′ which is        defined as

${{\overset{\rightarrow}{\alpha}}^{\prime}\overset{\Delta \;}{=}{\begin{bmatrix}\alpha_{1}^{\prime} \\\vdots \\\alpha_{N_{t}}^{\prime}\end{bmatrix}\overset{\Delta}{=}\begin{bmatrix}{\gamma_{1}\alpha_{1}} \\\vdots \\{\gamma_{N_{t}}\alpha_{N_{t}}}\end{bmatrix}}};$

-   -   

${{{\cdot \overset{\rightarrow}{\alpha}}\overset{\Delta}{=}\begin{bmatrix}\alpha_{1} \\\vdots \\\alpha_{N_{t}}\end{bmatrix}};} \equiv {N_{t} \times 1}$

is the signal vector which is defined by its m^(th) signal element,α_(m);

-   -   {right arrow over (ζ)}≡N×1 is the information vector which is        represented by the signal vector

$\begin{bmatrix}\alpha_{1} \\\vdots \\\alpha_{N_{t}}\end{bmatrix};$

-   -   χ{{right arrow over (α)}′} is the pre-coding function which        pre-codes the pre-weighted signal vector,

${{\overset{\rightarrow}{\alpha}}^{\prime}\overset{\Delta}{=}{\begin{bmatrix}\alpha_{1}^{\prime} \\\vdots \\\alpha_{N_{t}}^{\prime}\end{bmatrix}\overset{\Delta}{=}\begin{bmatrix}\begin{matrix}{\gamma_{1}\alpha_{1}} \\\vdots\end{matrix} \\{\gamma_{N_{t}}\alpha_{N_{t}}}\end{bmatrix}}},$

to produce the pre-coded and pre-weighted signal vector,

${{\overset{\rightarrow}{\alpha}}^{''}\overset{\Delta}{=}{\chi \left\{ {\overset{\rightarrow}{\alpha}}^{\prime} \right\}}};$

-   -   {right arrow over (γ)}≡N₁×1 is the pre-weighting vector defined        as

$\overset{\rightarrow}{\gamma}\overset{\Delta}{=}\begin{bmatrix}{\gamma_{1}\alpha_{1}} \\\vdots \\{\gamma_{N_{t}}\alpha_{N_{t}}}\end{bmatrix}$

which pre-weights the signal vector, {right arrow over (α)}, to producethe pre-weighted signal vector, {right arrow over (α)}′;

${\cdot {\overset{\rightarrow}{\beta}}^{w}}\overset{\Delta}{=}{\begin{bmatrix}\beta_{1}^{w} \\\vdots \\\beta_{N_{r}^{w}}^{w}\end{bmatrix} \equiv {N_{r}^{w} \times 1}}$

output of the w^(th) receiving device (terminal node) for 1≦w≦U_(r), oneelement per Rx;

${\cdot {\overset{\rightarrow}{\alpha}}^{''\; v}}\overset{\Delta}{=}{\begin{bmatrix}\alpha_{1}^{''\; v} \\\vdots \\\alpha_{N_{t}^{v}}^{''\; v}\end{bmatrix} \equiv {N_{t}^{v} \times 1}}$

input to the v^(th) transmitting device (access node) for 1≦v≦U_(t), oneelement per Tx;

${\cdot {\overset{\rightarrow}{\theta}}^{w}}\overset{\Delta}{=}{\begin{bmatrix}\theta_{1}^{w} \\\vdots \\\theta_{N_{r\;}^{w}}^{w}\end{bmatrix} \equiv {N_{r}^{w} \times 1}}$

noise contaminating the output of the w^(th) receiving device (terminalnode) for 1≦w≦U_(r), one element per Rx;

-   -   N_(t) ^(v) is the number of Txs in the v^(th) transmitting        device (access node) for 1≦v≦U_(t);    -   N_(r) ^(w) is the number of Rxs in the w^(th) receiving device        (terminal node) for 1≦w≦U_(r);

${\cdot h_{Ch}^{w}}\overset{\Delta}{=}{\begin{bmatrix}h_{Ch}^{w,1} & \ldots & h_{Ch}^{w,U_{t}}\end{bmatrix} \equiv {N_{r}^{w} \times N_{t}}}$

represents the communications Channel over which all U_(t) transmittingdevices (access nodes) transmit via their N_(t) Txs to the w^(th)receiving device (terminal node); and

${\cdot N_{t}}\overset{\Delta}{=}{N_{t}^{1} + \ldots + N_{t}^{U_{t}}}$

is the total number of Txs in the network.

Equations (34c) can be re-written to include the entire communicationsnetwork, i.e. to include all U_(r) receiving devices with all theirN_(r) Rxs as follows:

$\begin{matrix}{\begin{bmatrix}{\overset{\rightarrow}{\beta}}^{1} \\\vdots \\{\overset{\rightarrow}{\beta}}^{U_{r}}\end{bmatrix} = {{h_{Ch}\chi \left\{ {\overset{\rightarrow}{\alpha}}^{\prime} \right\}} + \overset{\rightarrow}{\theta}}} & \left( {36a} \right) \\{\mspace{65mu} {= {{h_{Ch}\chi \left\{ \begin{bmatrix}{\gamma_{1}\alpha_{1}} \\\vdots \\{\gamma_{N_{r}}\alpha_{N_{r}}}\end{bmatrix} \right\}} + \overset{\rightarrow}{\theta}}}} & \left( {36b} \right)\end{matrix}$

where

${\cdot h_{Ch}}\overset{\Delta}{=}{\begin{bmatrix}h_{Ch}^{1} \\\vdots \\h_{Ch}^{U_{r}}\end{bmatrix} = {\begin{bmatrix}h_{Ch}^{1,1} & \ldots & h_{Ch}^{1,U_{t}} \\\vdots & \ddots & \vdots \\h_{Ch}^{U_{r},1} & \ldots & h_{Ch}^{U_{r},U_{t}}\end{bmatrix} \equiv {N_{r} \times N_{t}}}}$

is referred to as the Channel matrix defined by its sub-matrix, h_(Ch)^(w,v), which is located at the w^(th) column block and at the v^(th)row block of h_(Ch);

-   -   sub-matrix h_(Ch) ^(w,v) connects the v^(th) transmitting device        (terminal node) to the w^(th) receiving device (access node) via        the various Txs in the v^(th) transmitting device (terminal        node) and the various Rxs in the w^(th) receiving device (access        node), for 1≦v≦U_(t), and 1≦w≦U_(r);    -   N_(r) ^(w) is the total number of received signal elements in        {right arrow over (β)}^(w), at the w^(th) receiving device        (terminal node) for 1≦w≦U_(r);    -   N_(t) ^(v) is the total number of transmitted pre-coded and        pre-weighted signal elements, {right arrow over (α)}″^(v), at        the v^(th) transmitting device (access node) for 1≦v≦U_(t);    -   N_(r) is the total number of received signal elements

$\quad\begin{bmatrix}{\overset{\rightarrow}{\beta}}^{1} \\\vdots \\{\overset{\rightarrow}{\beta}}^{U_{r}}\end{bmatrix}$

across all U_(r) receiving devices (access nodes);

-   -   N_(t) is the total number of transmitted pre-coded and        pre-weighted signal elements

$\quad\begin{bmatrix}{\overset{\rightarrow}{\alpha}}^{''\; 1} \\\vdots \\{\overset{\rightarrow}{\alpha}}^{''\; U_{t}}\end{bmatrix}$

across all U_(t) transmitting devices (terminal nodes);

${\cdot \begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix}}\overset{\Delta}{=}\begin{bmatrix}{\overset{\rightarrow}{\beta}}^{1} \\\vdots \\{\overset{\rightarrow}{\beta}}^{U_{r}}\end{bmatrix}$

where the U_(r) vectors,

$\begin{bmatrix}{\overset{\rightarrow}{\beta}}^{1} \\\vdots \\{\overset{\rightarrow}{\beta}}^{U_{r}}\end{bmatrix},$

are converted into N_(r) elements,

$\begin{bmatrix}\beta_{1} \\\vdots \\\beta_{N_{r}}\end{bmatrix};$

-   -   

${\bullet \begin{bmatrix}\alpha_{1}^{''} \\\vdots \\\alpha_{N_{t}}^{''}\end{bmatrix}}\overset{\Delta}{=}\begin{bmatrix}{\overset{\rightarrow}{\alpha}}^{''\; 1} \\\vdots \\{\overset{\rightarrow}{\alpha}}^{''\; U_{t}}\end{bmatrix}$

where the U_(t) vectors,

$\begin{bmatrix}{\overset{\rightarrow}{\alpha}}^{''\; 1} \\\vdots \\{\overset{\rightarrow}{\alpha}}^{''\; U_{t}}\end{bmatrix},$

are converted into N_(t) elements,

$\begin{bmatrix}\alpha_{1}^{''} \\\vdots \\\alpha_{N_{t}}^{''}\end{bmatrix};$

and

${\bullet \begin{bmatrix}\theta_{1} \\\vdots \\\theta_{N_{r}}\end{bmatrix}}\overset{\Delta}{=}\begin{bmatrix}{\overset{\rightarrow}{\theta}}^{1} \\\vdots \\{\overset{\rightarrow}{\theta}}^{U_{r}}\end{bmatrix}$

where the U_(r) vectors,

$\begin{bmatrix}{\overset{\rightarrow}{\theta}}^{1} \\\vdots \\{\overset{\rightarrow}{\theta}}^{U_{r}}\end{bmatrix},$

are converted into N_(r) elements,

$\begin{bmatrix}\theta_{1} \\\vdots \\\theta_{N_{r}}\end{bmatrix}.$

Note: When χ{•} is a linear pre-coding function, it reduces to a matrixx≡N_(t)×N_(r) multiplying {right arrow over (α)}′, i.e.

χ{{right arrow over (α)}′}=x{right arrow over (α)}′

DL aspects of the presently disclosed systems and methods intend toimprove the overall performance of the communication system representedby Equation (36b) by selecting the N_(r)×1 pre-weighting vector {rightarrow over (γ)}, corresponding to the N_(r) signal elements

$\quad\begin{bmatrix}\alpha_{1} \\\vdots \\\alpha_{N_{r}}\end{bmatrix}$

for a given pre-coding function χ{•} under a (minimum) Performanceconstraint and under the following transmit Power constraint:

∥χ{{right arrow over (α)}′}∥² ≦P   (37a)

where P is a fixed value, ∥χ{{right arrow over (α)}′}∥² is the

₂-norm of χ{{right arrow over (α)}′}. When χ{•} is a linear pre-codingfunction, Equation (37a) reduces to

|{x{right arrow over (α)}′} ₁|² + . . . +|{x{right arrow over (α)}′}_(N) _(t) |² ≦P   (37b)

where {x{right arrow over (α)}′}_(m) is the m^(th) element in the vectorx {right arrow over (α)}′.

The performance of a communications system may be improved in many ways,such as by increasing its overall bandwidth efficiency or by increasingits overall power efficiency. A compromise between both types ofefficiencies is to increase the sum rate, Σ_(w=1) ^(U) ^(r) Σ_(v=1) ^(U)^(t) Σ_(l=1) ^(N) ¹ ^(w) Σ_(m=1) ^(N) ¹ ^(v)

_(k,m) ^(w,v), of the system as defined in Equation (34). The DL aspectsof the presently disclosed systems and methods intend to increase thesum rate, Σ_(w=1) ^(U) ^(r) Σ_(v=1) ^(U) ^(t) Σ_(k=1) ^(N) ¹ ^(w)Σ_(m=1) ^(N) ¹ ^(v)

_(k,m) ^(w,v), for a Downlink (DL) MU-MIMO system by selecting theN_(t)×1 pre-weighting vector {right arrow over (γ)}, corresponding tothe N_(t) signal elements

$\quad\begin{bmatrix}\alpha_{1} \\\vdots \\\alpha_{N_{r}}\end{bmatrix}$

under a (maximum) transmit Power constraint and a (minimum) Performanceconstraint. Before showing the proposed pre-weighting selection methodfor the DL MU-MIMO, we review the general concept of Channel estimationin DL MU-MIMO and generally the various types of pre-coding that areavailable for MU-MIMO. Other pre-coding techniques may also be used.

Channel Estimation in DL MU-MIMO:

The full knowledge of the channel matrix, h_(Ch), at a transmitter issometimes referred to as Channel Side Full Information at Transmitter(CSFIT) which is defined as estimating at all U_(t) transmitting devices(access nodes) the communications channel matrix it_(Ch) between alltransmitting devices (access nodes) and all receiving devices (terminalnodes). This is possible as long as all U_(t) transmitting devices areable to cooperate, and the channel is reciprocal. Alternatively, if thechannel is slowly varying, then the transmitting devices can receive therequested channel estimates from the receiving devices. The partialknowledge of the channel matrix, h_(Ch) ^(w), at a transmitter issometimes referred to as Channel Side Partial Information at Transmitter(CSPIT) which is defined as estimating at the w^(th) receiving devicethe communications network matrix, h_(Ch) ^(w), between all transmittingdevices (access nodes) and the w^(th) receiving device (terminal node)and of the interference that is sensed by the w^(th) receiving device(terminal node).

Pre-Coding in DL MU-MIMO:

In addition to the Methods of Reception discussed earlier, there areseveral types of pre-coding that are available for DL MU-MIMO. The firsttype of pre-coding is intended to fully pre-compensate for the effectsof the channel at the transmitting devices (i.e. prior to transmission)in order to use a simple hard-decision detector at the receivingdevices. This type of pre-coding is sometimes referred to as IterativePre-Cancellation (IPC). It has been shown to represent a duality withSIC (e.g., as described with reference to process 700) at the receivingdevices. In other words, Method “A,” that was derived in the UL forselecting the ordering combination and the weighting values for ULMU-MIMO and which uses SIC at the receiving devices, has an equivalentmethod, Method “B,” for selecting the ordering combination and thepre-weighting values for DL MU-MIMO, including using IPC at thetransmitting devices. Often, the purpose of applying IPC pre-coding(e.g., as described with reference to step 1203) to the transmit signalin FIG. 12A is to pre-compensate for the effects of the channel bypre-diagonalizing the channel matrix.

The second type of pre-coding is intended to partially pre-compensatefor the effects of the channel at the transmitting devices (i.e. priorto transmission) and to rely on partial SIC at the receiving devices. Werefer to this type of pre-coding as partial IPC. Once again, partial IPCcombined with partial SIC have been shown to represent a duality witheither IPC at the transmitting devices or SIC at the receiving devices.Method “C” represents selecting the ordering combination and theweighting values for DL MU-MIMO using this type of pre-coding. Often,the purpose of applying partial IPC pre-coding (e.g., as described withreference to step 1213) to the transmit signal in FIG. 12B is topartially pre-compensate for the effects of the channel by pre-blockdiagonalizing the channel matrix.

Finally, the third type of pre-coding is intended to provide nopre-cancellation for the effects of the channel at the transmittingdevices (i.e. prior to transmission) and to rely instead on full SIC atthe receiving devices. Method “D” represents selecting the orderingcombination and the weighting values for DL MU-MIMO using this type ofpre-coding. Often, the purpose of applying no IPC pre-coding (e.g., asdescribed with reference to step 1223) to the transmit signal in FIG.12C is to ignore the effects of the channel by not pre-diagonalizing thechannel matrix. In each method, the terminal nodes may perform theirprocessing without communication with other terminal nodes.

There are generally two types of pre-coding: linear pre-coding, which isgenerally known to be sub-optimal and non-linear pre-coding which can beselected to be optimal (or asymptotically optimal).

Examples of Linear Pre-coding include:

-   -   a Zero Forcing (ZF) Pre-coding (or diagonalizing, or        null-steering): This is also generally referred to as Channel        inversion. It includes pre-compensating for the effect of the        channel using ZF pre-coding. For example, assuming in Equation        (35b) that h_(Ch) can be decomposed as

h_(Ch)=L_(Ch)Q_(Ch)   (38)

where

-   -   -   L_(Ch)≡N_(r)×N_(t) is a left triangular matrix; and        -   Q_(Ch)≡N_(t)×N_(t) is a unitary matrix.

    -   then x can be selected as

x=Q* _(Ch) L _(Ch) ⁻¹   (39)

-   -   so that h_(Ch)x=L_(Ch)Q_(Ch)Q*_(Ch)L_(Ch) ⁻¹ produces an        identity matrix. When h_(Ch) is ill-conditioned or close to        being singular, L_(Ch) ⁻¹ can be replaced by its pseudo-inverse,        h_(Ch) ^(pinv).    -   However, even in this case h_(Ch) ^(pinv) can suffer from a        power enhancement penalty similar to the noise enhancement        penalty in UL MU-MIMO. This type of pre-coding is generally        sub-optimal, but is suitable for a low noise (high SNR) high        interference environment.    -   Maximum Ratio Transmission (MRT) Pre-coding (or Maximum Ratio        Combining): It includes selecting x as h*_(Ch). This type of        pre-coding is generally suboptimal, but is suitable for a high        noise (low SNR) low interference environment. When the        interference between transmitting devices is high, the        performance of MRT pre-coding degrades rapidly. A compromise        between the severe degradation of the performance of MRT        pre-coding in an interference-dominated environment and that of        ZF pre-coding in a noise-dominated environment is Weighted        Minimum Mean Square Error (WMMSE) pre-coding.    -   Weighted Minimum Mean Square Error (WMMSE) Pre-coding (or        Weighted Wiener pre-filtering or regularized Zero Forcing        Pre-coding): This is also generally referred to as regularized        Channel inversion. It includes pre-compensating for the effect        of the channel using weighted MMSE pre-coding, i.e. replacing        L_(Ch) ⁻¹ in the ZF pre-coding by a function which is similar to        Equation (6) with pre-weighting, i.e. it takes into account the        effect of the noise and of the remaining interference. It        represents a compromise between the ZF pre-coding and the MRT        pre-coding.    -   Block Diagonalization: The method removes the interference from        signals intended to be received by each receiving device, the        interference being caused by signals intended for all other        U_(r)−1 receiving devices. This is in contrast to a        diagonalization method which removes the interference from        signals intended to be received by each Rx, the interference        being caused by signals intended for all other N_(r)−1 Rxs. The        distinction between diagonalizing and block-diagonalizing is due        to the fact that the w^(th) receiving device has N_(r) ^(w) Rxs,        hence it can pre-compensate for the N_(r) ^(w) received signals        using any number of pre-coding such as MRT pre-coding, ZF        pre-coding or WMMSE pre-coding.

Note: Linear pre-coding is generally sub-optimal due to the powerenhancement penalty, which exists on an ill-conditioned channel.Generally, non-linear pre-coding offers a performance improvement tolinear pre-coding at the cost of increased complexity in terms ofsearching for adequate non-linear pre-coding for a specific channel atthe transmitting devices, and removal of the effect of non-linearpre-coding at the receiving devices.

Non-Linear Pre-coding is generally referred to as Dirty Paper Pre-coding(DPC). It includes using a non-linear operation prior to invertingL_(Ch) to reduce the power enhancement penalty in the transmittingdevices, together with another non-linear operation in the receivingdevices to remove the effect of the first non-linear operation. Severaltypes of non-linear operations exist to reduce the power enhancementpenalty in the transmitting devices. They include addition ormultiplication at the transmitting devices of the information elementsby a set of pre-weighting values followed by a non-linear operationprior to transmission and another non-linear operation post reception.

Examples of non-linear pre-coding that may be used in IPC include:Costas Pre-coding; Tomlinson-Harashima Pre-coding; and VectorPerturbation:

Method “B” for Selecting Ordering and Pre-Weighting for DL MU-MIMO:

Assumptions “B”:

-   1. Since the portion of the network that Method B is concerned with    is a DL network, the method assumes that the U_(t) transmitting    devices (access nodes) are able to cooperate. This is a realistic    assumption when the transmitting devices are access nodes belonging    to the same network. Otherwise, the non-cooperating access nodes see    one another as separate interfering networks.-   2. Since the network is a DL network, the method assumes that the    channel matrix, h_(Ch), (or portions of the channel matrix such as    h_(Ch) ^(w)) is known by the U_(t) transmitting devices (access    nodes) and by the U_(r) receiving devices (terminal nodes). The    transmitting devices may know all parts of the channel matrix in an    LTE system, for example, by (a) the access nodes transmitting    reference signals, (b) the terminal nodes computing local channel    estimates, (c) the terminal nodes sending their local channel    estimates to the access nodes, and (d) the access nodes sending an    aggregated channel matrix to the terminal nodes using, for example,    a broadcast control channel such as PDCCH. It is realistic to assume    that the receiving devices are able to estimate their corresponding    channel, i.e. h_(Ch) ^(w). However, for the transmitting devices to    be able to estimate their corresponding channel requires the channel    to be either slowly varying in time or reciprocal.-   3. Based on the two previous assumptions that the channel matrix,    h_(Ch), (or portions of the channel matrix such as h_(Ch) ^(w)) is    known by the U_(t) transmitting devices and that the U_(t)    transmitting devices are able to cooperate, the method assumes that    full IPC is selected at the U_(t) transmitting devices (access    nodes) with pre-weighting of the signals prior transmission. This    assumption can be realistically carried out.-   4. Based on the previous assumption that full IPC is selected, the    method of reception is assumed to include a filter followed by a    hard-decision detector.

Constraints “B”:

-   1. When a minimum performance is required per receiving device    (terminal node), an equivalent desired received    Signal-to-Interference & Noise Ratio (SINR), η_(k,m) ^(w,v), for the    information element, ζ_(k,m) ^(w,v) between the m^(th) Tx of the    v^(th) transmitting device (access node) and the k^(th) Rx of the    w^(th) receiving device (terminal node), must be constrained to have    an equivalent lower bound, κ_(k,m) ^(w,v), for 1≦w≦U_(r), 1≦v≦U_(t),    1≦k≦N_(r) ^(w), 1≦m≦N_(t) ^(v)

η_(k,m) ^(w,v) ≧κ _(k,m) ^(w,v)   (40)

Equation (40) is referred to as the (minimum) received SINR constraint.An equivalent performance measure to the received SINR, η_(k,m) ^(w,v)is the bit Rate,

_(k,m) ^(w,v), since both are directly related as follows:

_(k, m)^(w, v) = _(k, m)^(w, v)log₂(1 + η_(k, m)^(w, v)).

The importance of such a constraint is to ensure a minimum downloadperformance for all terminal nodes. When IPC performs ideally, eachreceived SINR value reduces to a received SNR value, and the (minimum)received SINR constraint is replaced by a (minimum) received SNRconstraint.

-   2. Based on the received SINR constraint in Equation (40) for the    information element, ζ_(k,m) ^(w,v) between the m^(th) Tx of the    v^(th) transmitting device (access node) and the k^(th) Rx of the    w^(th) receiving device (terminal node), a pre-weighting vector,

${\overset{\rightarrow}{\gamma}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1} \\\vdots \\\gamma_{N_{t}}\end{bmatrix}},$

is selected such that the following transmit Power constraint inEquation (41) is met:

E{∥χ{{right arrow over (α)}′}∥²}≦P   (41)

where P is a pre-specified upper limit on the total transmitted powerand E{•} denotes statistical averaging with respect to the informationelements. The importance of such a constraint is to limit the averagetransmitted power for all terminal nodes.

Method “B”:

If a pre-weighting vector,

${\overset{\rightarrow}{\gamma}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1} \\\vdots \\\gamma_{N_{t}}\end{bmatrix}},$

is found which satisfies both the (minimum) received SINR constraint inEquation (40) and the (maximum) transmit Power constraint in Equation(41), then {right arrow over (γ)} is optimized such that the sum rateΣ_(w=1) ^(U) ^(r) Σ_(v=1) ^(U) ^(t) Σ_(k=1) ^(N) ^(r) ^(w) Σ_(m=1) ^(N)^(t) ^(v)

_(k,m) ^(w,v) is increased (or maximized). This can be accomplishedusing an “adjusted” waterfilling strategy. The “regular” waterfillingstrategy optimizes the sum rate with respect to {right arrow over (γ)}under the transmit Power constraint in Equation (41). The “adjusted”waterfilling strategy includes the received SINR constraint in Equation(40) together with the transmit Power constraint in Equation (41) whenoptimizing the sum rate. There are several ways to implement the“adjusted” waterfilling strategy. For example:

-   -   Way 1. The first way has two steps:        -   a. Find all ordering combinations and corresponding            pre-weighting values which satisfy the received SINR            constraint.        -   b. Select the ordering combination with the largest            corresponding sum rate using for example, the waterfilling            method subject to the transmit power constraint.    -   Way 2. The second way has two steps:        -   a. Find all ordering combinations and corresponding            pre-weighting values which satisfy the transmit power            constraint.        -   b. Select the ordering combination with the largest            corresponding sum rate using for example, the waterfilling            method subject to the received SINR constraint.    -   Way 3. The third way has two steps:        -   a. Find all ordering combinations and corresponding            pre-weighting values which satisfy both the received SINR            constraint and the transmit Power constraint.        -   b. Select the ordering combination with the largest            corresponding sum rate using for example two Lagrange            multipliers when obtaining the solution to the sum rate            optimization. The first Lagrange multiplier incorporates the            received SINR constraint, while the second Lagrange            multiplier incorporates the transmit power constraint.    -   Other ways to implement adjusted waterfilling strategies may        also be used. If no ordering combination is found which        satisfies the received SINR constraint, then use either Method V        or Method VI below for Relaxing Received SINR constraint in        Method “B.”

-   1. If a pre-weighting vector,

${\overset{\rightarrow}{\gamma}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1} \\\vdots \\\gamma_{N_{t}}\end{bmatrix}},$

which satisfies both the received SINR constraint in Equation (40) andthe transmit Power constraint in Equation (41), cannot be found, thenthe largest absolute value in {right arrow over (γ)} may be removed from{right arrow over (γ)} in Equations (40) and (41) and placed in a set,

_(o). This is repeated until both Equations (40) and (41) are satisfied.The pre-weighting elements in

_(o) are forced to take a zero value, i.e. their corresponding signalelements are not transmitted.

-   2. Alternatively, the received SINR constraint in Equation (40) can    be relaxed by reducing the constraint by a fixed factor, λ, for each    transmitting device (access node), until the transmit Power    constraint in Equation (41) is satisfied.-   3. All optimized N_(t) pre-weighting vectors (whether zero or    non-zero),

${\overset{\rightarrow}{\gamma}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1} \\\vdots \\\gamma_{N_{t}}\end{bmatrix}},$

are fed-back (e.g., as described with reference to step 609) by thetransmitting devices (access nodes) to all U_(r) receiving devices(terminal nodes).

The contributions of the presently disclosed systems and methods includethe selection of

$\overset{\rightarrow}{\gamma}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1} \\\vdots \\\gamma_{N_{t}}\end{bmatrix}$

for a DL MU-MIMO based on Method “B.”

Note: A difference between Method B for DL MU-MIMO and Method A for ULMU-MIMO is that for DL MU-MIMO, the complexity is mainly in thetransmitting devices (access nodes) while for UL MU-MIMO, the complexityis mainly in the receiving devices (access nodes), depending on whetherthe pre-coding is linear or non-linear.

Method V for Relaxing Received SINR Constraint in Method “B”:

Alternatively, if Equations (40) and (41) cannot be satisfied for anyordering of {right arrow over (γ)}, several remedies exist. For example,the maximum value, |γ_(S)|², which is obtained as

${{\gamma_{}}^{2}\overset{\Delta}{=}{\max \left\{ {{\gamma_{1}}^{2},{\gamma_{2}}^{2},\ldots \mspace{14mu},{\gamma_{N_{t\;}}}^{2}} \right\}}},$

is removed from Equations (40) and (41), and placed in a set, {rightarrow over (γ)}_(S). These are repeated until both Equations (40) and(41) are satisfied. The formed set of squared values in {right arrowover (γ)}_(S)is then replaced by i a zero value, i.e. the signalelements that correspond to {right arrow over (γ)}_(S)are nottransmitted.

Method VI for Relaxing Received SINR Constraint in Method “B”:

Another remedy for the case when Equations (40) and (41) cannot besatisfied for any ordering of {right arrow over (γ)}, is to reduceEquation (40) by a factor, γ, which would allow for an orderingcombination to of y i be found which satisfies both Equations (40) and(41). In other words, instead of accommodating only a few of theterminal nodes as in the previous strategy, this strategy attempts toaccommodate all terminal nodes in a fair fashion.

Note: When choosing between the two Methods for relaxing the receivedSINR constraint, one can rely on the variance of {|γ₁|², |γ₂|², . . . ,|γ_(N) _(t) |²}. If the variance is larger than a pre-specifiedthreshold, then Method V is selected, otherwise, Method VI is selected.

Method “C” for Selecting Ordering and Pre-Weighting for DL MU-MIMO:

Assumptions “C”:

-   1. Since the network is a DL network, the method assumes that the    U_(t) transmitting devices (access nodes) are able to cooperate.    This is a realistic assumption when the transmitting devices are    access nodes belonging to the same network. Otherwise, the    non-cooperating access nodes see one another as two separate    interfering networks.-   2. Since the network is a DL network, the method assumes that the    channel matrix, h_(Ch), (or portions of the channel matrix such as    h_(Ch) ^(w)) is known by the U_(t) transmitting devices (access    nodes) and by the U_(r) receiving devices (terminal nodes). It is    realistic to assume that the receiving devices are able to estimate    their corresponding channel, i.e. h_(Ch) ^(w). However, for the    transmitting devices to be able to estimate their corresponding    channel requires the channel to be either slowly varying in time or    reciprocal.-   3. Based on the two previous assumptions that the channel matrix,    h_(Ch), (or portions of the channel matrix such as h_(Ch) ^(w)) is    known by the U_(t) transmitting devices and that the U_(t)    transmitting devices are able to cooperate, the method assumes that    partial IPC is selected at the U_(t) transmitting devices (access    nodes) with pre-weighting of the signals prior to transmission. This    assumption can be realistically carried out using for example    Q*_(Ch) as the pre-coding matrix, which forces the channel matrix    h_(Ch) to be a left triangular matrix L_(Ch). Other pre-coding    matrices may also be used.-   4. Based on the previous assumption that partial IPC is selected,    the method of reception is assumed to includes a filter across the    antennas (Rxs) that belong to a single receiving device, followed by    a partial SIC detector to remove the effects of the remaining    portion of the channel. For example, using Q*_(Ch) as the pre-coding    matrix, forces the channel matrix h_(Ch) to be a left triangular    matrix L_(Ch). In order to remove the effects of the left triangular    matrix L_(Ch), a partial SIC detector is required. In this specific    case, the first ordered Tx (terminal node antenna), which    corresponds to the top diagonal element in L_(Ch), contains ideally    no interference. The second ordered Tx contains the effect of the    first element, and so on.-   5. Based on the previous two assumptions, the method assumes that    the ordering combination of all N_(t) signal elements is based on    maximizing the sum rate. Since the selection of the pre-weighting    elements affects their corresponding sum rate, the ordering    combination of all N_(t) signal elements is carried out    simultaneously with the selection of the pre-weighting elements.

Constraints “C”:

-   1. When a minimum performance is required per receiving device    (terminal node), an equivalent desired received    Signal-to-Interference & Noise Ratio (SINR), η₁ _(k,m,o) , for the    m^(th) ordered signal element, α₁ _(k,m,o) , at the k^(th) device    must be constrained to have a lower bound, κ₁ _(k,m,o) , for    1≦m≦N_(t):

η₁ _(k,m,o) ≧κ ₁ _(k,m,o)   (42)

Equation (42) is referred to as the received SINR constraint. Theimportance of such a constraint is to ensure no error propagation in thepartial SIC detector.

-   2. Based on the received SINR constraint in Equation (42) for the    m^(th) ordered information element, α₁ _(k,m,o) , a pre-weighting    vector,

${{\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}},$

is selected such that the following transmit Power constraint is met:

E{∥χ{{right arrow over (α)}′}∥²}≦P   (43)

where P is a pre-specified upper limit on the total transmitted power,E{•} denotes statistical averaging with respect to the signal vector{right arrow over (α)}. The importance of such a constraint is to limitthe average transmitted power for all terminal nodes.

Method “C”:

If a pre-weighting vector,

${{\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}},$

is found which satisfies both the received SINR constraint in Equation(42) and the transmit Power constraint in Equation (43), then optimize{right arrow over (γ)}_(o) such that the sum rate, Σ_(w=1) ^(U) ^(r)Σ_(v=1) ^(U) ^(t) Σ_(k=1) ^(N) ^(r) ^(w) Σ_(m=1) ^(N) ^(t) ^(v)

_(k,m) ^(w,v) in Equation (34) is increased (or maximized). This can beaccomplished using an “adjusted” waterfilling strategy. The “regular”waterfilling strategy optimizes the sum rate with respect to {rightarrow over (γ)}_(o) under the transmit Power constraint in Equation(43). The “adjusted” waterfilling strategy includes the received SINRconstraint in Equation (42) together with the transmit Power constraintin Equation (43) when optimizing the sum rate. There are several ways toimplement the “adjusted” waterfilling strategy. For example:

-   -   Way 1. The first way has two steps:        -   a. Find all ordering combinations and corresponding            pre-weighting values which satisfy the received SINR            constraint.        -   b. Select the ordering combination with the largest            corresponding sum rate using for example, the waterfilling            method subject to the transmit power constraint.    -   Way 2. The second way has two steps:        -   a. Find all ordering combinations and corresponding            pre-weighting values which satisfy the transmit power            constraint.        -   b. Select the ordering combination with the largest            corresponding sum rate using for example, the waterfilling            method subject to the received SINR constraint.    -   Way 3. The third way has two steps:        -   a. Find all ordering combinations and corresponding            pre-weighting values which satisfy both the received SINR            constraint and the transmit Power constraint.        -   b. Select the ordering combination with the largest            corresponding sum rate using for example two Lagrange            multipliers when obtaining the solution to the sum rate            optimization. The first Lagrange multiplier incorporates the            received SINR constraint, while the second Lagrange            multiplier incorporates the transmit power constraint.    -   Other ways to implement adjusted waterfilling strategies may        also be used. If no ordering combination is found which        satisfies the received SINR constraint, then use either Method        VII, VIII, IX or X below for Relaxing Received SINR constraint        in Method “C.”

-   1. If more than one ordering combination for the pre-weighting    vector,

${{\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{r},o}}\end{bmatrix}},$

are found which satisfy both the received SINR constraint in Equation(42) and the transmit Power constraint in Equation (43), then optimize{right arrow over (γ)}_(o) for all ordering combinations such that thesum rate Σ_(w=1) ^(U) ^(r) Σ_(v=1) ^(U) ^(t) Σ_(k=1) ^(N) ^(r) ^(w)Σ_(m=1) ^(N) ^(t) ^(v)

_(k,m) ^(w,v) in Equation (34) is maximized for each orderingcombination, and select the ordering combination, which corresponds tothe largest sum rate.

-   2. If a pre-weighting vector,

${{\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}},$

which satisfies both the received SINR constraint in Equation (42) andthe transmit Power constraint in Equation (43), cannot be found, thenthe largest absolute value in {right arrow over (γ)}_(o) is removed from{right arrow over (γ)}_(o) in Equations (42) and (43) and placed in aset,

_(o). This is repeated until both Equations (42) and (43) are satisfied.The pre-weighting elements in

_(o) are forced to take a zero value, i.e. their corresponding signalelements are not transmitted.

-   3. Alternatively, the received SINR constraint in Equation (42) can    be relaxed by reducing the constraint by a fixed factor, λ, for each    transmitting device (access nodes), until the transmit

Power constraint in Equation (43) is satisfied.

-   4. All optimized pre-weighting vectors (whether zero or non-zero),

${{\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}},$

are fed-back (e.g., as described with reference to step 609) by thetransmitting devices (access nodes) to all U_(r) receiving devices(terminal nodes) with their corresponding order. The contributions ofthe presently disclosed systems and methods include the selection of

${{\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}},$

and its ordering combination, for a DL MU-MIMO based on Method “C.”

Two major differences exist between Method “C” for DL MU-MIMO and Method“A” for UL MU-MIMO:

-   -   Pre-weighting and pre-coding are both required for DL MU-MIMO in        Method “C,” while for UL MU-MIMO only pre-weighting is        performed. This is explained as follows. The number, N_(t), of        Txs is usually larger than the number of Rxs per receiving        device (terminal node) in DL MU-MIMO. For this reason, the        pre-weighting and pre-coding functions are performed across all        N_(t) Txs. In other words, instead of selecting a scalar        pre-weighting element γ_(m) for the m^(th) information element        and a filter at the Rxs as in UL MU-MIMO, we combine in DL        MU-MIMO the pre-weighting scalar with a filter to produce a        pre-weighting and pre coding (pre-filter) vector with N_(t)        output elements for the m^(th) signal element for 1≦m≦N_(t),        i.e. one element for each one of the N_(t) Txs;    -   Another difference between Method “C” for DL MU-MIMO and Method        “A” for UL MU-MIMO is that for DL MU-MIMO, the complexity is        mainly shared between the transmitting devices (access nodes)        and the receiving devices (terminal nodes) while for UL MU-MIMO,        the complexity is mainly in the receiving devices (access        nodes).

An Embodiment for Pre-Weighting Selection in Method “C”:

Assume that the received SINR constraint in Equation (42) is written asη₁ _(m,o) ≧κ₁ _(m,o) and that Q*_(Ch) is used as the pre-coding matrixin the transmitting devices where Q*_(Ch) is defined in Equation (38) aspart of the LQ decomposition of h_(Ch), i.e. h_(Ch)=L_(Ch)Q_(Ch). Inthis case, the remaining effect from the channel is the left triangularmatrix L_(Ch), which can be removed using a partial SIC at the receivingdevices. This approach is supported by the fact that the combination ofpartial IPC (and pre-weighting) at the transmitting devices and partialSIC at the receiving devices for DL MU-MIMO is equivalent topre-weighting at the transmitting devices and full SIC at the receivingdevices for UL MU-MIMO. In this case, an equivalent set of received SINRequations can be derived after filtering the received signals at thew^(th) receiving device (terminal node) N_(r) ^(w) Rxs. At the i^(th)iteration, we have

$\begin{matrix}{\eta_{1_{i,o}}^{w}\overset{\Delta}{=}{\frac{E\left\{ {{\hat{\alpha}}_{1_{i,o}}^{w\; \prime}}^{2} \right\}}{{E\left\{ {{h_{{Est}_{1_{i,o}}}^{w}\begin{bmatrix}\theta_{1_{i,o}}^{w} \\\vdots \\\theta_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix}}}^{2} \right\}} + {E\left\{ {{\sum_{l = 2}^{N_{r}^{w\; \prime} - i + 1}{\hat{\alpha}}_{l_{i,o}}^{w\; ''}}}^{2} \right\}}} \geq \kappa_{1_{i,o}}^{w}}} & (44)\end{matrix}$

for 1≦i≦N_(r) ^(w)′ and for 1≦w≦U_(r), where

-   -   N_(r) ^(w)′ is the number of interfering signals out of the        total N_(t) signal elements, which remain in the received signal        at the w^(th) Rx due to the left triangular matrix L_(Ch);    -   η₁ _(i,o) ^(w) is the ordered received SINR for the first        information element to be detected in the i^(th) iteration at        the w^(th) receiving device (terminal node) using N_(r) ^(w) Rxs        and its corresponding minimum required received SINR, κ₁ _(i,o)        ^(w);

$\cdot \begin{matrix}{{\hat{\alpha}}_{1_{i,o}}^{w\; {\prime\prime}}\overset{\Delta}{=}{h_{{Est}_{1_{i,o}}}^{w}\begin{bmatrix}\beta_{1_{i,o}}^{w} \\\vdots \\\beta_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix}}} \\{= {{h_{{Est}_{1_{i,o}}}^{w}h_{{Ch}_{i,o}}^{w}{\overset{\rightarrow}{\alpha}}_{i,o}^{w\; ''}} + {h_{{Est}_{1_{i,o}}}^{w}\begin{bmatrix}\theta_{1_{i,o}}^{w} \\\vdots \\{\theta \;}_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix}}}} \\{= {{h_{{Est}_{1_{i,o}}}^{W}h_{{Ch}_{i,o}}^{w}{x_{i,o}^{w}\begin{bmatrix}{\gamma_{1_{i,o}}^{w}\alpha_{1_{i,o}}^{w}} \\0 \\\vdots \\0\end{bmatrix}}} + {h_{{Est}_{1_{i,o}}}^{w}\begin{bmatrix}\theta_{1_{i,o}}^{w} \\\vdots \\\theta_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix}}}}\end{matrix}$

is the estimated value of the 1^(st) pre-coded and pre-weighted signalelement, α₁ _(i,o) ^(w)″, in {right arrow over (α)}_(i,o) ^(w)″ afterfiltering the received signal elements

$\quad\begin{bmatrix}\beta_{1_{i,o}}^{w} \\\vdots \\\beta_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix}$

with a row vector,

h_(Est_(1_(i, o)))^(w),

at the w^(th) receiving device (terminal node);

h_(Est_(1_(i, o)))^(w) ≡ 1 × N_(r)^(w)

is the 1^(st) row vector of the estimation matrix h_(Est) _(i,o) ^(w)which is used estimate the 1^(st) pre-coded and pre-weighted signalelement, α₁ _(i,o) ^(w)″, in {right arrow over (α)}_(i,o) ^(w)″; and

$\begin{bmatrix}\beta_{1_{i,o}}^{w} \\\vdots \\\beta_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix} \equiv {N_{r}^{w} \times 1}$

is a vector consisting of the received signals elements at the w^(th)receiving device (terminal node), which remain after removing theeffects of the (i−1) previously detected information elements;

-   -   x_(i,o) ^(w) is the ordered pre-coding matrix for the i^(th)        iteration at the w^(th) receiving device (terminal node);

${\hat{\alpha}}_{l_{i,o}}^{w\; ''}\overset{\Delta}{=}{h_{{Est}_{1_{i,o}}}^{w}h_{{Ch}_{i,o}}^{w}{x_{i,o}^{w}\begin{bmatrix}0 \\\vdots \\0 \\{\gamma_{l_{i,o}}^{w}\alpha_{l_{i,o}}^{w}} \\0 \\\vdots \\0\end{bmatrix}}}$

the interference component corresponding to the l^(th) element, α_(l)_(i,o) ^(w)″, in {right arrow over (α)}_(i,o) ^(w)″ after filtering thereceived signal elements

$\quad\begin{bmatrix}\beta_{1_{i,o}}^{w} \\\vdots \\\beta_{N_{r_{i,o}}}^{w}\end{bmatrix}$

with the row vector,

h_(Est_(1_(i, o)))^(w)

where 2≦l≦N_(r) ^(w)′−i+1; and

${h_{{Est}_{1_{i,o}}}^{w}\begin{bmatrix}\theta_{1_{i,o}}^{w} \\\vdots \\\theta_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix}}\quad$

is the noise component that results from filtering the noise vector

$\begin{bmatrix}\theta_{1_{i,o}}^{w} \\\vdots \\\theta_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix}\quad$

with the row vector

h_(Est_(1_(i, o)))^(w).

The relationship between the estimate, {right arrow over (α)}₁ _(i,o)^(w)″, of the pre-weighted signal element α₁ _(i,o) ^(w)″ and thedetected information element, {hacek over (ζ)}₁ _(i,o) ^(w), of theinformation element ζ₁ _(i,o) ^(w) is that {circumflex over (α)}₁ _(i,o)^(w)″ must be divided by an a priori known factor, then a hard-decisiondetector is applied to its inverse to remove the 1:1 function whichconverts information element ζ₁ _(i,o) ^(w) to signal element α₁ _(i,o)^(w), to obtain detected information element ζ₁ _(i,o) ^(w). The effectof information element ζ₁ _(i,o) ^(w) on the received signal elements

$\begin{bmatrix}\beta_{1_{i,o}} \\\vdots \\\beta_{N_{r_{i,o}}}\end{bmatrix}\quad$

is then removed using detected information element {hacek over (ζ)}₁_(i,o) ^(w) assuming no error propagation. In order to justify such anassumption, the pre-weighting vector, {right arrow over (γ)}_(o), mustbe selected to satisfy the (minimum) received SINR constraint inEquation (42) under the (maximum) Power constraint in Equation (43).However, since the signal vector, {right arrow over (α)}, is unknown tothe receiving devices, then two assumptions are made in Equation (44):

-   -   the elements of the signal vector, {right arrow over (α)}, are        independent identically distributed (iid) with zero mean and        variance σ_(α) ² and    -   the elements of the noise vector, {right arrow over (θ)}, are        independent identically distributed (iid) with zero mean and        variance σ_(θ) ².

In other words, Equation (44) can re-written as

$\begin{matrix}{\mspace{79mu} {{\eta_{1_{i,o}}^{w}\overset{\Delta}{=}{\frac{\sigma_{\alpha}^{2}{\xi_{1_{i,o}}^{w}}^{2}}{{\sigma_{\theta}^{2}h_{{Est}_{1_{i,o}}}^{w}h_{{Est}_{1_{i,o}}}^{w*}} + {\sigma_{\alpha}^{2}{\sum\limits_{l = 2}^{N_{r}^{w^{\prime}} - i + 1}{\xi_{1_{i,o}}^{w}}^{2}}}} \geq \kappa_{i,o}^{w}}}\mspace{79mu} {where}}} & (45) \\{\begin{matrix}{{\xi_{1_{i,o}}^{w}}^{2}\overset{\Delta}{=}{h_{{Est}_{1_{i,o}}}^{w}h_{{Ch}_{i,o}}^{w}x_{i,o}^{w}{\gamma_{1_{i,o}}^{w}\begin{bmatrix}1 & 0 & \ldots & 0 \\0 & 0 & \ldots & 0 \\\vdots & \vdots & \ddots & \vdots \\0 & 0 & \ldots & 0\end{bmatrix}}\gamma_{1_{i,o}}^{w*}x_{i,o}^{w*}h_{{Ch}_{i,o}}^{w*}h_{{Est}_{1_{i,o}}}^{w*}}} \\{= {{{h_{{Est}_{1_{i,o}}}^{w}h_{{Ch}_{i,o}}^{w}\left\{ x_{i,o}^{w} \right\}_{1}\gamma_{1_{i,o}}^{w}}}^{2}\left( {46b} \right)}}\end{matrix}\mspace{79mu} {and}} & \left( {46a} \right) \\{\begin{matrix}{{\xi_{l_{i,o}}^{w}}^{2}\overset{\Delta}{=}{h_{{Est}_{1_{i,o}}}^{w}h_{{Ch}_{i,o}}^{w}x_{i,o}^{w}{\gamma_{l_{i,o}}^{w}\begin{bmatrix}0 & \ldots & 0 & 0 & 0 & \ldots & 0 \\\vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\0 & \ldots & 0 & 0 & 0 & \ldots & 0 \\0 & \vdots & \vdots & 1 & \vdots & \vdots & \vdots \\0 & \ldots & 0 & 0 & 0 & \ldots & 0 \\\vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\0 & \ldots & 0 & 0 & 0 & \ldots & 0\end{bmatrix}}}} \\{{\gamma_{l_{i,o}}^{w*}x_{i,o}^{w*}h_{{Ch}_{i,o}}^{w*}h_{{Est}_{1_{i,o}}}^{w*}}} \\{= {{{h_{{Est}_{1_{i,o}}}^{w}h_{{Ch}_{i,o}}^{w}\left\{ x_{i,o}^{w} \right\}_{l}\gamma_{1_{i,o}}^{w}}}^{2}\left( {47b} \right)}}\end{matrix}\quad} & \left( {47a} \right)\end{matrix}$

where {x_(i,o) ^(w)}_(l) is the l^(th) column vector in x_(i,o) ^(w). Ifthe estimation filter,

h_(Est_(1_(i, o)))^(w),

is normalized, i.e.

h_(Est_(1_(i, o)))^(w)h_(Est_(1_(i, o)))^(w*) = 1,

then {x_(i,o) ^(w)}_(l)γ₁ _(i,o) ^(w)

$\begin{matrix}{\begin{matrix}{\eta_{1_{i,o}}^{w}\overset{\Delta}{=}{\frac{\sigma_{\alpha}^{2}{\xi_{1_{i,o}}^{w}}^{2}}{\sigma_{\theta}^{2} + {\sigma_{\alpha}^{2}{\sum\limits_{l = 2}^{N_{r}^{w^{\prime}} - i + 1}{\xi_{1_{i,o}}^{w}}^{2}}}} \geq \kappa_{1_{i,o}}^{w}}} \\{= {\frac{{{h_{{Est}_{1_{i,o}}}^{w}h_{{Ch}_{i,o}}^{w}\left\{ x_{i,o}^{w} \right\}_{1}\gamma_{1_{i,o}}^{w}}}^{2}}{\frac{1}{v} + {\sum\limits_{l = 2}^{N_{r}^{w^{\prime}} - i + 1}{{h_{{Est}_{1_{i,o}}}^{w}h_{{Ch}_{i,o}}^{w}\left\{ x_{i,o}^{w} \right\}_{l}\gamma_{1_{i,o}}^{w}}}^{2}}} \geq {\kappa_{1_{i,o}}^{w}\left( {48b} \right)}}}\end{matrix}\quad} & \left( {48a} \right)\end{matrix}$

In this case, the (maximum) Power constraint can be re-written as

$\begin{matrix}{{E\left\{ {{\chi \left\{ {\overset{\rightarrow}{\alpha}}^{\prime} \right\}}}^{2} \right\}} = {{\sigma_{\alpha}^{2}\left\{ {{\gamma_{1_{1,o}}{^{2}{{+ \ldots} +}}\gamma_{1_{N_{t},o}}}}^{2} \right\}} \leq P}} & (49)\end{matrix}$

If the ordered pre-weighting vector, {right arrow over (γ)}_(o), isfound to satisfy both the SINR constraint in Equations (43) and thePower constraint in Equation (44), then the next step is to optimize thesum rate.

This can be accomplished using an “adjusted” waterfilling strategy.Other strategies may also be used.

A Solution of Equation (48) in Method “C”:

When i=N_(r) ^(w)′, Equation (48) reduces to

${\frac{\sigma_{\alpha}^{2}{\xi_{1_{N_{r}^{w^{\prime}},o}}^{w}}^{2}}{\sigma_{\theta}^{2}} \geq \kappa_{1_{N_{r}^{w^{\prime}},o}}^{w}},$

or equivalently

$\begin{matrix}{{\xi_{1_{N_{r}^{w^{\prime}},o}}^{w}}^{2} \geq \frac{\kappa_{1_{N_{r}^{w^{\prime}},o}}^{w}}{v}} & \left( {50a} \right)\end{matrix}$

where

$v\overset{\Delta}{=}{\sigma_{\theta}^{2}/\sigma_{a}^{2}}$

is the normalized SNR corresponding to the transmitted signal elements.

When i=N_(r) ^(w)′−1, Equation (48) reduces to

${\frac{\sigma_{\alpha}^{2}{\xi_{1_{{N_{r}^{w^{\prime}} - 1},o}}^{w}}^{2}}{\sigma_{\theta}^{2} + {\sigma_{\alpha}^{2}{\xi_{2_{{N_{r}^{w^{\prime}} - 1},o}}^{w}}^{2}}} \geq \kappa_{1_{{N_{r}^{w^{\prime}} - 1},o}}^{w}},$

or equivalently

$\begin{matrix}{{\xi_{1_{{N_{r}^{w^{\prime}} - 1},o}}^{w}}^{2} \geq {\kappa_{1_{{N_{r}^{w^{\prime}} - 1},o}}^{w}\left( {\frac{1}{v} + {\xi_{2_{{N_{r}^{w^{\prime}} - 1},o}}^{w}}^{2}} \right)}} & \left( {50b} \right)\end{matrix}$

When i=N_(r) ^(w)′−2, Equation (48) reduces to

${\frac{\left. \sigma_{\alpha}^{2} \middle| \xi_{1_{{N_{r}^{w\; \prime} - 2},o}}^{w} \right|^{2}}{\left. {\sigma_{\theta}^{2} + \sigma_{\alpha}^{2}} \middle| \xi_{2_{{N_{r}^{w\; \prime} - 2},o}}^{w} \middle| {}_{2}{+ \sigma_{\alpha}^{2}} \middle| \xi_{3_{{N_{r}^{w\; \prime} - 2},o}}^{w} \right|^{2}} \geq \kappa_{1_{{N_{t} - 2},o}}^{w}},$

or equivalently

$\begin{matrix}\left| \xi_{1_{{N_{r}^{w\; \prime} - 2},o}}^{w} \middle| {}_{2}{\geq {\kappa_{1_{{N_{r}^{w\; \prime} - 2},o}}^{w}\left( {\frac{1}{v} +} \middle| \xi_{2_{{N_{r}^{w\; \prime} - 2},o}}^{w} \middle| {}_{2}{+ \left| \xi_{3_{{N_{r}^{w\; \prime} - 2},o}}^{w} \right|^{2}} \right)}} \right. & \left( {50c} \right)\end{matrix}$

In general, at the i^(th) iteration, we have

$\begin{matrix}\left| \xi_{1_{i,o}}^{w} \middle| {}_{2}{\geq {\kappa_{1_{i,o}}^{w}\left( {\frac{1}{v} +} \middle| \xi_{2_{i,o}}^{w} \middle| {}_{2}{+ \left| \xi_{3_{i,o}}^{w} \middle| {}_{2}{{+ \cdots} +} \middle| \xi_{N_{r}^{w\; \prime} - i + 1_{i,o}}^{w} \right|^{2}} \right)}} \right. & \left( {50d} \right)\end{matrix}$

for 1≦i≦N_(r) ^(w)′. From Equation (51a),

γ_(1_(N_(r)^(w ′), o))^(w)

can be derived by solving

$\begin{matrix}\left| {h_{{Est}_{1_{N_{r}^{w\; \prime},o}}}^{w}h_{{Ch}_{N_{r}^{w\; \prime},o}}^{w}\left\{ x_{N_{r}^{w\; \prime},o}^{w} \right\}_{1}\gamma_{1_{N_{r}^{w\; \prime},o}}^{w}} \middle| {}_{2}{\geq \frac{{{}_{}^{\kappa 1w}{}_{}^{w\; \prime}},o}{v}} \right. & \left( {51a} \right)\end{matrix}$

From Equation (50b),

γ_(1_(N_(r)^(w ′) − 1, o))^(w)

can be derived by solving

$\begin{matrix}\left| {h_{{Est}_{1_{{N_{r}^{w\; \prime} - 1},o}}}^{w}h_{{Ch}_{{N_{r}^{w\; \prime} - 1},o}}^{w}\left\{ x_{{N_{r}^{w\; \prime} - 1},o}^{w} \right\}_{1}\gamma_{1_{{N_{r}^{w\; \prime} - 1},o}}^{w}} \middle| {}_{2}{\geq {\kappa_{1_{{N_{r}^{w\; \prime} - 1},o}}^{w}\left( \left. {\frac{1}{v} +} \middle| \xi_{2_{{N_{r}^{w\; \prime} - 1},o}}^{w} \right|^{2} \right)}} \right. & \left( {51b} \right)\end{matrix}$

In general, from Equation (50d), γ₁ _(i,o) ^(w) can be derived bysolving

$\begin{matrix}\left| {h_{{Est}_{1_{i,o}}}^{w}h_{{Ch}_{i,o}}^{w}\left\{ x_{i,o}^{w} \right\}_{1}\gamma_{1_{i,o}}^{w}} \middle| {}_{2}{\geq {\kappa_{1_{i,o}}^{w}\left( {\frac{1}{v} +} \middle| \xi_{2_{i,o}}^{w} \middle| {}_{2}{+ \left| \xi_{3_{i,o}}^{w} \middle| {}_{2}{{+ \cdots} +} \middle| \xi_{N_{r}^{w\; \prime} - i + 1_{i,o}}^{w} \right|^{2}} \right)}} \right. & (52)\end{matrix}$

After deriving the pre-weighting vector,

${{\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}},$

the transmit Power constraint in Equation (49) is tested as follows

$\begin{matrix}{\left\{ \left| \gamma_{1_{i,o}} \middle| {}_{2}{{+ \cdots} +} \middle| \gamma_{1_{N_{t},o}} \right|^{2} \right\} \leq \frac{P}{\sigma_{\alpha}^{2}}} & (53)\end{matrix}$

A Selection of the Ordering of

$\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{t}}\end{bmatrix}\mspace{14mu} {{into}\mspace{14mu}\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{t},o}}^{w}\end{bmatrix}}$

in Method “C”:

Based on Equations (52) and the transmit Power constraint in Equation(53), the ordering of

$\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{t}}\end{bmatrix}\mspace{14mu} {{into}\mspace{14mu}\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{t},o}}^{w}\end{bmatrix}}$

is based on selecting

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

in such a way as to maximize the sum rate.

One possible way for ordering

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{t},o}}^{w}\end{bmatrix}\quad$

and for selecting

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

which satisfies the constraints in Equations (52) and (53), is toexhaustively search for all possible ordering combinations of

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

until at least one ordering combination satisfies Equations (52) and(53). If one or more ordering combinations satisfy Equations (52) and(53), then the next step is to optimize the sum rate using an “adjusted”waterfilling strategy, and to select the ordering combination whichoffers the largest sum rate.

Another possible way for ordering

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{t},o}}^{w}\end{bmatrix}\quad$

and for selecting

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

which satisfies the constraints in Equations (52) and (53), is to sortfrom

$\begin{bmatrix}\frac{\kappa_{1}^{w}}{\left. ||{h_{{Est}_{1_{1,o}}}^{w}h_{{Ch}_{1,o}}^{w}\left\{ x_{i\; 1\; o}^{w} \right\}_{1}} \right.||^{2}} \\\vdots \\\frac{\kappa_{N_{r}}^{w}}{\left. ||{h_{{Est}_{1_{N_{r},o}}h_{{Ch}_{N_{r},o}}^{w}}^{w}\left\{ x_{N_{r},o}^{w} \right\}_{1}} \right.||^{2}}\end{bmatrix}\quad$

high to low for 1≦w≦U_(r), where

-   -   h_(Est) _(m) ^(w)≡1×N_(r) ^(w) is tje m^(th) row vector of the        estimation matrix h_(Est) ^(w) at the w^(th) receiving device        (terminal node), which is used to estimate the m^(th) signal        element, α″_(m), in {right arrow over (α)}″; and    -   h_(Ch) ^(w)≡N_(r) ^(w)×N₁ Channel between the N_(t) Txs and the        w^(th) receiving device (terminal node).

Given that there are U_(r) receiving devices, sorting

$\begin{bmatrix}\frac{\kappa_{1}^{w}}{\left. ||{h_{{Est}_{1_{1,o}}}^{w}h_{{Ch}_{1,o}}^{w}\left\{ x_{i\; 1\; o}^{w} \right\}_{1}} \right.||^{2}} \\\vdots \\\frac{\kappa_{N_{r}}^{w}}{\left. ||{h_{{Est}_{1_{N_{r},o}}h_{{Ch}_{N_{r},o}}^{w}}^{w}\left\{ x_{N_{r},o}^{w} \right\}_{1}} \right.||^{2}}\end{bmatrix}{\quad\quad}$

from high to low can result in more than one (up to U_(r)) uniqueordering combination, which satisfy Equations (52) and (53). In thiscase, the next step is to optimize the sum rate using an “adjusted”waterfilling strategy for each ordering combination, and to select theordering combination that offers the largest sum rate.

Alternatively, if Equations (52) and (53) cannot be satisfied for anyordering of

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{r},o}}^{w}\end{bmatrix},$

several remedies exist:

Method VII for Relaxing received SINR constraint in Method “C”:

For example, the maximum value, |γ₁ _(S,o) |², which is obtained as

${\left| \gamma_{1_{,o}} \right|^{2}\overset{\Delta}{=}{\max \left\{ {\left| \gamma_{1_{1,o}} \right|^{2},\left| \gamma_{1_{2,o}} \right|^{2},\ldots,\left| \gamma_{1_{{N_{t} - 1},o}} \right|^{2}} \right\}}},$

is removed from Equations (52) and (53), and placed in a set, {rightarrow over (γ)}_(S) _(o) , and its corresponding indices, S is placed inanother set, S_(o). These are repeated until both Equations (52) and(53) are satisfied. The formed set of squared values in {right arrowover (γ)}_(S) _(o) , is then replaced by a zero value, i.e. thecorresponding information elements that correspond to {right arrow over(γ)}_(S) _(o) are not transmitted.

Method VIII for Relaxing Received SINR Constraint in Method “C”:

Another remedy for the case when Equations (52) and (53) cannot besatisfied for any ordering combination of

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{r},o}}^{w}\end{bmatrix},$

is to reduce Equations (52) by a factor which would allow for anordering combination to be found which satisfies both Equations (52) and(53). In other words, instead of accommodating only a few of theterminal nodes as in the previous strategy, this strategy attempts toaccommodate all terminal nodes in a fair fashion.

Note: When choosing between the two Methods for relaxing the SINRconstraint, one can rely on the variance of {|γ₁ _(1,o) |², |γ₁ _(2,0)|², . . . , |γ₁ _(Nt−1,o) |²}. If the variance is above a pre-specifiedthreshold, then Method VII is selected, otherwise, Method VIII isselected.

Another Embodiment of Pre-Weighting Selection in Method “C”:

For the special case of UL MU-MIMO where N_(r) ^(w)=1, for all values of1≦w≦U_(r), i.e. each receiving device (terminal node) has only oneantenna, DL MU-MIMO reduces to DL MU-MISO. In this case, the set ofordered SINR equations in Equation (49b) is written as:

$\begin{matrix}{\eta_{1_{i,o}}^{w} = {\frac{\left. ||{h_{{Ch}_{i,o}}^{w}\left\{ x_{i,o}^{w} \right\}_{1}\gamma_{1_{i,o}}^{w}} \right.||^{2}}{\left. {\frac{1}{v} + \sum_{l = 2}^{N_{r}^{w\; \prime} - i + 1}} \middle| {h_{{Ch}_{i,o}}^{w}\left\{ x_{i,o}^{w} \right\}_{l}\gamma_{1_{i,o}}^{w}} \right|^{2}} \geq \kappa_{1_{i,o}}^{w}}} & (54)\end{matrix}$

Another Selection of the Ordering of

$\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{r}}\end{bmatrix}\mspace{14mu} {{into}\mspace{14mu}\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{r},o}}^{w}\end{bmatrix}}$

in Method “C”

Based on Equation (54) and the Power constraint in Equation (53), theordering of

$\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{r}}\end{bmatrix}\quad$

into

$\quad\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{r},o}}^{w}\end{bmatrix}$

is based on selecting

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

in such a way as to maximize the sum rate.

One possible way for ordering

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{r},o}}^{w}\end{bmatrix}\quad$

and for selecting

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

which satisfies the constraints in Equations (53) and (54), is toexhaustively search for all possible ordering combinations of

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

until at least one ordering combination is found which satisfiesEquations (53) and (54). If one or more ordering combinations are foundwhich satisfy Equations (53) and (54), then the next step is to optimizethe sum rate using an “adjusted” waterfilling strategy, and to selectthe ordering combination, which offers the largest sum rate.

Another possible way for ordering

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{r},o}}^{w}\end{bmatrix}\quad$

and for selecting

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{i,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

which satisfies the constraints in Equations (53) and (54), is to sort

$\begin{bmatrix}\frac{\kappa_{1}^{w}}{\left. ||{h_{{Ch}_{1,o}}^{w}\left\{ x_{i\; 1\; o}^{w} \right\}_{1}} \right.||^{2}} \\\vdots \\\frac{\kappa_{N_{r}}^{w}}{\left. ||{h_{{Ch}_{N_{r},o}}^{w}\left\{ x_{N_{r},o}^{w} \right\}_{1}} \right.||^{2}}\end{bmatrix}\quad$

from high to low for 1≦w≦U_(r).

Given that there are U_(r) receiving devices, sorting

$\begin{bmatrix}\frac{\kappa_{1}^{w}}{\left. ||{h_{{Ch}_{1,o}}^{w}\left\{ x_{i\; 1\; o}^{w} \right\}_{1}} \right.||^{2}} \\\vdots \\\frac{\kappa_{N_{r}}^{w}}{\left. ||{h_{{Ch}_{N_{r},o}}^{w}\left\{ x_{N_{r},o}^{w} \right\}_{1}} \right.||^{2}}\end{bmatrix}\quad$

from high to low can result in more than one (up to U_(r)) uniqueordering which satisfy Equations (53) and (54). In this case, the nextstep is to optimize the sum rate using an “adjusted” waterfillingstrategy for each ordering, and to select the ordering combination thatoffers the largest sum rate.

Method IX for Relaxing received SINR constraint in Method “C”:

Alternatively, if Equations (53) and (54) cannot be satisfied for anyordering of

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{r},o}}^{w}\end{bmatrix}{\quad,}$

several remedies exist. For example, the maximum value, |γ₁ _(S,o) |²,which is obtained as

${\left| \gamma_{1_{,o}} \right|^{2}\overset{\Delta}{=}{\max \left\{ {\left| \gamma_{1_{1,o}} \right|^{2},\left| \gamma_{1_{2,o}} \right|^{2},\ldots,\left| \gamma_{1_{{N_{t} - 1},o}} \right|^{2}} \right\}}},$

is removed from Equations (53) and (54), and placed in a set, {rightarrow over (γ)}_(S) _(o) , and its corresponding indices, S is placed inanother set, S_(o). These are repeated until both Equations (53) and(54) are satisfied. The formed set of squared values in {right arrowover (γ)}_(S) _(o) , is then replaced by a zero value, i.e. thecorresponding information elements that correspond to {right arrow over(γ)}_(S) _(o) are not transmitted.

Method X for Relaxing Received SINR Constraint in Method “C”:

Another remedy for the case when Equations (53) and (54) cannot besatisfied for any ordering of

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{r},o}}^{w}\end{bmatrix},$

is to reduce Equation (53) by a factor which would allow for an orderingcombination to be found which satisfies both Equation (53) and (54). Inother words, instead of accommodating only a few of the terminal nodesas in the previous strategy, this strategy attempts to accommodate allterminal nodes in a fair fashion.Note: When choosing between the two Methods for relaxing the SINRconstraint, one can rely on the variance of {|γ₁ _(1,o) |², |γ₁ _(2,o)|², . . . , |γ₁ _(Nt−1,o) |²}. If the variance is above a pre-specifiedthreshold, then Method IX is selected, otherwise, Method X is selected.

Method “D” for Selecting Ordering and Pre-Weighting for DL MU-MIMO:

Assumptions “D”:

-   1. Since the network is a DL network, the method assumes that the    U_(t) transmitting devices (access nodes) are able to cooperate.    This is a realistic assumption when the transmitting devices are    access nodes belonging to the same network. Otherwise, the    non-cooperating access nodes see one another as separate interfering    networks.-   2. Since the network is a DL network, the method assumes that the    channel matrix, h_(Ch), (or portions of the channel matrix such as    h_(Ch) ^(w)) is known by the U_(t) transmitting devices (access    nodes) and by the U_(r) receiving devices (terminal nodes). It is    realistic to assume that the receiving devices are able to estimate    their corresponding channel, i.e. h_(Ch) ^(w). However, for the    transmitting devices to be able to estimate their corresponding    channel requires the channel to be either slowly varying in time or    reciprocal.-   3. Based on the two previous assumptions that the channel matrix,    h_(Ch), (or portions of the channel matrix such as h_(Ch) ^(w)) is    known by the U_(t) transmitting devices and that the U_(t)    transmitting devices are able to cooperate, the method assumes that    no pre-coding is selected at the U_(t) transmitting devices (access    nodes) with pre-weighting of the signals prior transmission. This    assumption can be realistically carried out.-   4. Based on the previous assumption that no pre-coding is selected,    the method of reception is assumed to include a filter followed by a    full SIC detector.-   5. Based on the previous two assumptions, the method assumes that    the ordering of all N_(t) signal elements is based on their    corresponding received Signal-to-Noise & Interference Ratio (SINR),    η_(k,m), ordered into η₁ _(k,m,o) , from high to low, where the    interference at the i^(th) iteration is due to the N_(t)−i remaining    signal elements. Since the selection of the pre-weighting elements    affects their corresponding received SINR, the ordering of all N_(t)    information elements based on their corresponding received SINR is    carried out simultaneously with the selection of the pre-weighting    elements.

Constraints “D”:

-   1. When a minimum performance is required per receiving device    (terminal node), an equivalent desired received    Signal-to-Interference & Noise Ratio (SINR), η₁ _(k,m,o) , for the    m^(th) ordered signal element, α₁ _(m,o) , at the k^(th) Tx, must be    constrained to have a lower bound, κ₁ _(k,m,o) , for 1≦m≦N_(t) and    or 1≦k≦N_(r):

η₁ _(k,m,o) ≧κ ₁ _(k,m,o)   (55)

Equation (55) is referred to as the (minimum) received SINR constraint.The importance of such a constraint is to ensure no error propagationfrom the SIC.

-   2. Based on the received SINR constraint in Equation (55) for the    m^(th) ordered information element, α₁ _(m,o) , a pre-weighting    vector,

${{\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}},$

is selected such that the following Power constraint is met:

E{∥{right arrow over (α)}′∥ ² }≦P   (56)

where P is a pre-specified upper limit on the total transmitted powerand E{•} denotes statistical averaging with respect to the informationelements. The importance of such a constraint is to limit the averagetransmitted power for all terminal nodes.

Method “D”:

If a pre-weighting vector,

${{\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}},$

is found which satisfies both the SINR constraint in Equation (55) andthe Power constraint in Equation (56), then the method optimizes {rightarrow over (γ)}_(o) such that the sum rate in Equation (34) is increased(or maximized). This can be accomplished using an “adjusted”waterfilling strategy. The “regular” waterfilling strategy optimizes thesum rate with respect to {right arrow over (γ)}_(o) under the Powerconstraint in Equation (56). The “adjusted” waterfilling strategyincludes the SINR constraint in Equation (55) together with the Powerconstraint in Equation (56) when optimizing the sum rate. There areseveral ways to implement the “adjusted” waterfilling strategy. Forexample:

-   -   Way 1. The first way has two steps:        -   a. Find all ordering combinations and corresponding            pre-weighting values which satisfy the received SINR            constraint.        -   b. Select the ordering combination with the largest            corresponding sum rate using for example, the waterfilling            method subject to the transmit power constraint.    -   Way 2. The second way has two steps:        -   a. Find all ordering combinations and corresponding            pre-weighting values which satisfy the transmit power            constraint.        -   b. Select the ordering combination with the largest            corresponding sum rate using for example, the waterfilling            method subject to the received SINR constraint.    -   Way 3. The third way has two steps:        -   a. Find all ordering combinations and corresponding            pre-weighting values which satisfy both the received SINR            constraint and the transmit Power constraint.        -   b. Select the ordering combination with the largest            corresponding sum rate using for example two Lagrange            multipliers when obtaining the solution to the sum rate            optimization. The first Lagrange multiplier incorporates the            received SINR constraint, while the second Lagrange            multiplier incorporates the transmit power constraint.    -   Other ways to implement adjusted waterfilling strategies may        also be used. If no ordering combination is found which        satisfies the received SINR constraint, then use either Method        X1, XII, XIII or XIV below for Relaxing Received SINR constraint        in Method “D.”

-   1. If more than one ordering for the pre-weighting vector,

${{\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}},$

are found which satisfy both the SINR constraint in Equation (55) andthe Power constraint in Equation (56), then {right arrow over (γ)}_(o)is optimized for all ordering combinations such that the sum rate ismaximized for each ordering combination. The ordering combination, whichcorresponds to the largest sum rate is selected.

-   2. If a pre-weighting vector,

${{\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}},$

which satisfies both the received SINR constraint in Equation (55) andthe transmit Power constraint in Equation (56), cannot be found then thelargest absolute value in {right arrow over (γ)}_(o) is removed from{right arrow over (γ)}_(o) in Equations (55) and (56) is i and placed ina set,

_(o). This is repeated until both Equations (55) and (56) are satisfied.The pre-weighting elements in

_(o) are forced to take a zero value, i.e. their correspondinginformation elements are not transmitted.

-   3. Alternatively, the received SINR constraint in Equation (55) can    be relaxed by reducing the pre-weighting elements by a fixed factor,    λ, for each transmitting device (access nodes), until the Power    constraint in Equation (56) is satisfied.-   4. All optimized N_(t) pre-weighting vectors (whether zero or    non-zero),

${{\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}},$

are fed-back (e.g., as described with reference to step 609) by thereceiving devices (access nodes) to all U_(r) receiving devices(terminal nodes) with their corresponding order.

The contributions of the presently disclosed systems and methods includethe selection of

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

and its ordering combination, for a DL MU-MIMO based on Method “D.”

An Embodiment for Pre-Weighting Selection in Method “D”:

Assume that the received SINR constraint in Equation (55) is written asη₁ _(k,m,o) ≧κ₁ _(k,m,o) .

Since IPC and pre-weighting at the transmitting devices for DL MU-MIMOare equivalent to pre-weighting at the transmitting devices and SIC atthe receiving devices for UL MU-MIMO, then SIC is adopted here in thepresent embodiment for selecting ordering and pre-weighting. In otherwords, an equivalent set of received SINR equations can be derived afterfiltering the received signals at the w^(th) receiving device (terminalnode) using N_(r) ^(w) Rxs. At the i^(th) iteration, we have

$\begin{matrix}{\eta_{1_{i,o}}^{w}\overset{\Delta}{=}{\frac{E\left\{ \left| {\hat{\alpha}}_{1_{i,o}}^{w\; \prime} \right|^{2} \right\}}{{E\left\{ \left| {h_{{Est}_{1_{i,o}}}^{w}\begin{bmatrix}\theta_{1_{i,o}}^{w} \\\vdots \\\theta_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix}} \right|^{2} \right\}} + {E\left\{ \left| {\sum_{l = 2}^{N_{t} - i + 1}{\hat{\alpha}}_{l_{i,o}}^{w\; \prime}} \right|^{2} \right\}}} \geq \kappa_{1_{i,o}}^{w}}} & (57)\end{matrix}$

for 1≦i≦N_(r) for 1<w<U_(r), where

-   -   η₁ _(i,o) ^(w) is the ordered received SINR for the first        information element to be detected in the i^(th) iteration at        the w^(th) receiving device (terminal node) using N_(r) ^(w) Rxs        and its corresponding minimum required SINR, κ₁ _(i,o) ^(w);

$\begin{matrix}{{\bullet \mspace{14mu} {\hat{\alpha}}_{1_{i,o}}^{w\; \prime}}\overset{\Delta}{=}{h_{{Est}_{1_{i,o}}}^{w}\begin{bmatrix}\beta_{1_{i,o}}^{w} \\\vdots \\\beta_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix}}} \\{{= {{h_{{Est}_{1_{i,o}}}^{w}h_{{Ch}_{i,o}}^{w}{\overset{\rightarrow}{\alpha}\;}_{i,o}^{w\; \prime}} + {h_{{Est}_{i,o}}^{w}\begin{bmatrix}\theta_{1_{i,o}}^{w} \\\vdots \\\theta_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix}}}}} \\{{= {{h_{{Est}_{1_{i,o}}}^{w}{h_{{Ch}_{i,o}}^{w}\begin{bmatrix}{\gamma_{1_{i,o}}^{w}\alpha_{1_{i,o}}^{w}} \\0 \\\vdots \\0\end{bmatrix}}} + {h_{{Est}_{1_{i,o}}}^{w}\begin{bmatrix}\theta_{1_{i,o}}^{w} \\\vdots \\\theta_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix}}}}}\end{matrix}$

is the estimated value of the 1^(st) pre-weighted signal element, α₁_(i,o) ^(w)′, after filtering the received signal elements

$\begin{bmatrix}\beta_{1_{i,o}}^{w} \\\vdots \\\beta_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix}\quad$

with a row vector,

h_(Est_(1_(i, o)))^(w),

at the w^(th) receiving device (terminal node);

•  h_(Est_(1_(i, o)))^(w) ≡ 1 × N_(r)^(w)

is the 1^(st) row vector of the estimation matrix h_(Est) _(i,o) ^(w)which is used to estimate the 1^(st) pre-weighted signal element, α₁_(i,o) ^(w)′, and

${\bullet \mspace{14mu}\begin{bmatrix}\beta_{1_{i,o}}^{w} \\\vdots \\\beta_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix}} \equiv {N_{r}^{w} \times 1}$

is a vector consisting of the received signal elements at the w^(th)receiving device (terminal node), which remain after removing theeffects of the (i−1) previously detected information element;

-   -   {right arrow over (α)}_(i,o) ^(w)′ is the ordered pre-weighted        signal vector for the i^(th) iteration at the w^(th) receiving        device (terminal node);

${\bullet \mspace{14mu} {\hat{\alpha}}_{l_{i,o}}^{w\; \prime}}\overset{\Delta}{=}{= {h_{{Est}_{1_{i,o}}}^{w}{h_{{Ch}_{i,o}}^{w}\mspace{14mu}\begin{bmatrix}{\mspace{70mu} 0} \\{\mspace{76mu} \vdots} \\{\mspace{70mu} 0} \\{\gamma_{l_{i,o}}^{w}\alpha_{l_{i,o}}^{w}} \\{\mspace{70mu} 0} \\{\mspace{76mu} \vdots} \\{\mspace{70mu} 0}\end{bmatrix}}}}$

is the interference component corresponding to the l^(th) element, α_(l)_(i,o) ^(w)′, in {right arrow over (α)}_(i,o) ^(w)′ after filtering thereceived signal elements

$\begin{bmatrix}\beta_{1_{i,o}} \\\vdots \\\beta_{N_{r_{i,o}}}\end{bmatrix}\quad$

with the row vector,

h_(Est_(1_(i, o)))^(w)

where 2≦l≦N_(r)−i+1; and

$\bullet \mspace{14mu} {h_{{Est}_{1_{i,o}}}^{w}\mspace{14mu}\begin{bmatrix}\theta_{1_{i,o}}^{w} \\\vdots \\\theta_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix}}$

is the noise component that results from filtering the noise vector

$\begin{bmatrix}\theta_{1_{i,o}}^{w} \\\vdots \\\theta_{N_{r_{i,o}}^{w}}^{w}\end{bmatrix}\quad$

with the row vector

h_(Est_(1_(i, o)))^(w).

The relationship between the estimate, {circumflex over (α)}₁ _(i,o)^(w)′, of the pre-weighted signal element α₁ _(i,o) ^(w)′ and thedetected information element, {hacek over (ζ)}₁ _(i,o) ^(w), of theinformation element ζ₁ _(i,o) ^(w) is that estimated pre-weighted signalelement {circumflex over (α)}₁ _(i,o) ^(w)′ must be divided by an apriori known factor, then a reverse operation to the 1:1 function whichconverts information element ζ₁ _(i,o) ^(w) to signal element α₁ _(i,o)^(w) must be undertaken. For example, when the 1:1 function whichconverts information element

₁ _(i,o) ^(w) to signal element α₁ _(i,o) ^(w) is an encoder, thereverse operation is a decoder; when the 1:1 function which convertsinformation element ζ₁ _(i,o) ^(w) to signal element α₁ _(i,o) ^(w) is ascrambler, the reverse operation is a de-scrambler; and when the 1:1function which converts information element

₁ _(i,o) ^(w) to signal element α₁ _(i,o) ^(w) is an interleaver, thereverse operation is a de-interleaver.

Finally, a hard-decision detector is applied to {circumflex over (ζ)}₁_(i,o) ^(w) to obtain detected information element {hacek over (ζ)}₁_(i,o) ^(w). The effect of information element ζ₁ _(i,o) ^(w) on thereceived signal elements

$\begin{bmatrix}\beta_{1_{i,o}} \\\vdots \\\beta_{N_{r_{i,o}}}\end{bmatrix}\quad$

is then removed using detected information element {hacek over (ζ)}₁_(i,o) ^(w) assuming no error propagation. In order to justify such anassumption, the pre-weighting vector, {right arrow over (γ)}_(o), mustbe selected to satisfy the (minimum) received SINR constraint inEquation (55) under the (maximum) transmit Power constraint in Equation(56). However, since the signal vector, {right arrow over (α)}, isunknown to the receiving devices, then two assumptions are made inEquation (57):

-   -   the elements of the signal vector, {right arrow over (α)}, are        independent identically distributed (iid) with zero mean and        variance σ_(α) ² and    -   the elements of the noise vector are independent identically        distributed (iid) with zero mean and variance σ_(θ) ².

In other words, Equation (39) can re-written as

$\begin{matrix}{{\eta_{1_{i,o}}^{w}\overset{\Delta}{=}{\frac{\left. \sigma_{\alpha}^{2} \middle| \xi_{1_{i,o}}^{w} \right|^{2}}{\left. {{\sigma_{\theta}^{2}h_{{Est}_{1_{i,o}}}^{w}h_{{Est}_{1_{i,o}}}^{w*}} + {\sigma_{\alpha}^{2}\sum_{l = 2}^{N_{t} - i + 1}}} \middle| \xi_{l_{i,o}}^{w} \right|^{2}} \geq \kappa_{1_{i,o}}^{w}}}{where}} & (58) \\{\left| \xi_{1_{i,o}}^{w} \right|^{2}\overset{\Delta}{=}{h_{{Est}_{1_{i,o}}}^{w}h_{{Ch}_{i,o}}^{w}{\gamma_{1_{i,o}}^{w}\begin{bmatrix}1 & 0 & \cdots & 0 \\0 & 0 & \cdots & 0 \\\vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & 0\end{bmatrix}}\gamma_{1_{i,o}}^{w*}h_{{Ch}_{i,o}}^{w*}h_{{Est}_{1_{i,o}}}^{w*}}} & \left( {59a} \right) \\{= \left| {\left\{ {h_{{Est}_{1_{i,o}}}h_{{Ch}_{i,o}}} \right\}_{1}\gamma_{1_{i,o}}^{2}} \middle| {}_{2}{and} \right.} & \left( {59b} \right) \\{\left| \xi_{1_{i,o}}^{w} \right|^{2}\overset{\Delta}{=}{h_{{Est}_{1_{i,o}}}^{w}h_{{Ch}_{i,o}}^{w}{\gamma_{l_{i,o}}^{w}\begin{bmatrix}0 & \cdots & 0 & 0 & 0 & \cdots & 0 \\\vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\0 & \cdots & 0 & 0 & 0 & \cdots & 0 \\0 & \vdots & \vdots & 1 & \vdots & \vdots & \vdots \\0 & \cdots & 0 & 0 & 0 & \cdots & 0 \\\vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\0 & \cdots & 0 & 0 & 0 & \cdots & 0\end{bmatrix}}\gamma_{1_{i,o}}^{w*}h_{{Ch}_{i,o}}^{w*}h_{{Est}_{1_{i,o}}}^{w*}}} & \left( {60a} \right) \\{= \left| {\left\{ {h_{{Est}_{1_{i,o}}}h_{{Ch}_{i,o}}} \right\}_{1}\gamma_{1_{i,o}}^{2}} \right|^{2}} & \left( {60b} \right)\end{matrix}$

where

{h_(Est_(1_(i, o)))h_(Ch_(i, o))}_(l)

is the l^(th) element in

h_(Est_(1_(i, o)))h_(Ch_(i, o)).

It the estimation filter,

h_(Est_(1_(i, o)))^(w),

is normalized, i.e.

h_(Est_(1_(i, o)))^(w)h_(Est_(1_(i, o)))^(w*) = 1,

then

$\begin{matrix}{\eta_{1_{i,o}}^{w}\overset{\Delta}{=}{\frac{\left. \sigma_{\alpha}^{2} \middle| \xi_{1_{i,o}}^{w} \right|^{2}}{\left. {\sigma_{\theta}^{2} + {\sigma_{\alpha}^{2}\sum_{l = 2}^{N_{t} - i + 1}}} \middle| \xi_{l_{i,o}}^{w} \right|^{2}} \geq \kappa_{1_{i,o}}^{w}}} & \left( {61a} \right) \\{= {\frac{\left| {\left\{ {h_{{Est}_{1_{i,o}}}h_{{Ch}_{i,o}}} \right\}_{1}\gamma_{1_{i,o}}^{w}} \right|^{2}}{\left. {\frac{1}{v} + \sum_{l = 2}^{N_{t} - i + 1}} \middle| {\left\{ {h_{{Est}_{1_{i,o}}}h_{{Ch}_{i,o}}} \right\}_{l}\gamma_{l_{i,o}}^{w}} \right|^{2}} \geq \kappa_{1_{i,o}}^{w}}} & \left( {61b} \right)\end{matrix}$

In this case, the Power constraint can be re-written as

E{∥{right arrow over (α)}′∥ ²}=σ_(α) ²{|γ₁ _(1,o) |²+ . . . +|γ₁ _(Nt,o)|² }≦P   (62)

If the ordered pre-weighting vector, {right arrow over (γ)}_(o), isfound to satisfy both the SINR constraint in Equations (61) and thePower constraint in Equation (62), then the next step is to optimize thesum rate in Equation (18). This can be accomplished using an “adjusted”waterfilling strategy.

A Solution of Equation (61b) in Method “D”:

When i=N_(t), Equation (61b) reduces to

${\frac{\left. \sigma_{\alpha}^{2} \middle| \xi_{1_{N_{t},o}}^{w} \right|^{2}}{\sigma_{\theta}^{2}} \geq \kappa_{1_{N_{t},o}}^{w}},$

or equivalently

$\begin{matrix}\left| \xi_{1_{N_{t},o}}^{w} \middle| {}_{2}{\geq \frac{\kappa_{1_{N_{t},o}}^{w}}{v}} \right. & \left( {63a} \right)\end{matrix}$

where

$v\overset{\Delta}{=}{\sigma_{\theta}^{2}\text{/}\sigma_{\alpha}^{2}}$

is the SNR corresponding to the transmitted signal elements.

When i=N_(t)−1, Equation (61b) reduces to

${\frac{\left. \sigma_{\alpha}^{2} \middle| \xi_{1_{{N_{t} - 1},o}}^{w} \right|^{2}}{\left. {\sigma_{\theta}^{2} + \sigma_{\alpha}^{2}} \middle| \xi_{2_{{N_{t} - 1},o}}^{w} \right|^{2}} \geq \kappa_{1_{{N_{t} - 1},o}}^{w}},$

or equivalently

$\begin{matrix}\left| \xi_{1_{{N_{t} - 1},o}}^{w} \middle| {}_{2}{\geq {\kappa_{1_{{N_{t} - 1},o}}^{w}\left( \left. {\frac{1}{v} +} \middle| \xi_{2_{{N_{t} - 1},o}}^{w} \right|^{2} \right)}} \right. & \left( {63b} \right)\end{matrix}$

When i=N_(t)−2, Equation (43) reduces to

${\frac{\left. \sigma_{\alpha}^{2} \middle| \xi_{1_{{N_{t} - 2},o}}^{w} \right|^{2}}{\left. {\sigma_{\theta}^{2} + \sigma_{\alpha}^{2}} \middle| \xi_{2_{{N_{t} - 2},o}}^{w} \middle| {}_{2}{+ \sigma_{\alpha}^{2}} \middle| \xi_{3_{{N_{t} - 2},o}}^{w} \right|^{2}} \geq \kappa_{1_{{N_{t} - 2},o}}^{w}},$

or equivalently

$\begin{matrix}\left| \xi_{1_{{N_{t} - 2},o}}^{w} \middle| {}_{2}{\geq {\kappa_{1_{{N_{t} - 2},o}}^{w}\left( {\frac{1}{v} +} \middle| \xi_{2_{{N_{t} - 2},o}}^{w} \middle| {}_{2}{+ \left| \xi_{3_{{N_{t} - 2},o}}^{w} \right|^{2}} \right)}} \right. & \left( {63c} \right)\end{matrix}$

In general, at the i^(th) iteration, we have

$\begin{matrix}\left| \xi_{1_{i,o}}^{w} \middle| {}_{2}{\geq {\kappa_{1_{i,o}}^{w}\left( {\frac{1}{v} +} \middle| \xi_{2_{i,o}}^{w} \middle| {}_{2}{+ \left| \xi_{3_{i,o}}^{w} \middle| {}_{2}{{+ \cdots} +} \middle| \xi_{N_{t} - i + 1_{i,o}}^{w} \right|^{2}} \right)}} \right. & \left( {63d} \right)\end{matrix}$

for 1≦i≦N_(t). From Equation (63a),

γ_(1_(N_(r), o))

can be derived by solving

$\begin{matrix}\left| {\left\{ {h_{{Est}_{1_{N_{t},o}}}h_{{Ch}_{N_{t},o}}} \right\}_{1}\gamma_{1_{N_{t},o}}^{w}} \middle| {}_{2}{\geq \frac{\kappa_{1_{N_{t},o}}^{w}}{v}} \right. & \left( {64a} \right)\end{matrix}$

From Equation (63b), γ₁ _(Nt−1,o) can be derived by solving

$\begin{matrix}\left| {\left\{ {h_{{Est}_{1_{{N_{t} - 1},o}}}h_{{Ch}_{{N_{t} - 1},o}}} \right\}_{1}\gamma_{1_{{N_{t} - 1},o}}^{w}} \middle| {}_{2}{\geq {\kappa_{1_{{N_{t} - 1},o}}^{w}\left( \left. {\frac{1}{v} +} \middle| \xi_{2_{{N_{t} - 1},o}}^{w} \right|^{2} \right)}} \right. & \left( {64b} \right)\end{matrix}$

In general, from Equation (63d), γ₁ _(i,o) ^(w) can be derived bysolving

$\begin{matrix}\left| {\left\{ {h_{{Est}_{1_{i,o}}}h_{{Ch}_{i,o}}} \right\}_{1}\gamma_{1_{i,o}}^{w}} \middle| {}_{2}{\geq {\kappa_{1_{i,o}}^{w}\left( {\frac{1}{v} +} \middle| \xi_{2_{i,o}}^{w} \middle| {}_{2}{+ \left| \xi_{3_{i,o}}^{w} \middle| {}_{2}{{+ \cdots} +} \middle| \xi_{N_{t} - i + 1_{i,o}}^{w} \right|^{2}} \right)}} \right. & \left( {64d} \right)\end{matrix}$

After deriving the pre-weighting vector,

${{\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}},$

the Power constraint in Equation (62) is tested as follows

$\begin{matrix}\left| \gamma_{1_{i,o}} \middle| {}_{2}{{+ \cdots} +} \middle| \gamma_{1_{N_{t},o}} \middle| {}_{2}{\leq \frac{P}{\sigma_{\alpha}^{2}}} \right. & (65)\end{matrix}$

A Selection of the Ordering of

$\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{r}}\end{bmatrix}\mspace{14mu} {{into}\mspace{14mu}\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{t},o}}^{w}\end{bmatrix}}$

in Method “D”:

Based on Equations (64) and the Power constraint in Equation (65), theordering of

$\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{t}}\end{bmatrix}\mspace{14mu} {{into}\mspace{14mu}\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{t},o}}^{w}\end{bmatrix}}$

is based on selecting

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

in such a way as to maximize the sum rate. One possible way for ordering

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{t},o}}^{w}\end{bmatrix}{\quad\quad}$

and for selecting

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

which satisfies the constraints in Equations (64) and (65), is toexhaustively search for all possible ordering combinations of

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

until at least one ordering combination satisfies Equations (64) and(65). If one or more ordering combinations satisfy Equations (64) and(65), then the next step is to optimize the sum rate using an “adjusted”waterfilling strategy, and to select the ordering combination whichoffers the largest sum rate.

Another possible way for ordering

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{t},o}}^{w}\end{bmatrix}{\quad\quad}$

and for selecting

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

which satisfies the constraints in Equations (64) and (65), is to sort

$\begin{bmatrix}\frac{\kappa_{1}}{\left. ||{h_{{Est}_{1}}^{w}h_{Ch}^{w}} \right.||^{2}} \\\vdots \\\frac{\kappa_{N_{t}}}{\left. ||{h_{{Est}_{N_{t}}}^{w}h_{Ch}^{w}} \right.||^{2}}\end{bmatrix}\quad$

from high to low for 1≦w≦U_(r), where

-   -   h_(Est) _(m) ^(w)≡1×N_(r) ^(w) is the m^(th) row vector of the        estimation matrix h_(Est) ^(w) at the w^(th) receiving device        (terminal node), which is used to estimate the m^(th)        pre-weighted signal element, α′_(m), in {right arrow over (α)}′;        and    -   h_(Ch) ^(w)≡N_(r) ^(w)×N_(t) Channel between the N_(t) Txs and        the w^(th) receiving device (terminal node).

Given that there are U_(r) receiving devices, sorting

$\begin{bmatrix}\frac{\kappa_{1}}{\left. ||{h_{{Est}_{1}}^{w}h_{Ch}^{w}} \right.||^{2}} \\\vdots \\\frac{\kappa_{N_{t}}}{\left. ||{h_{{Est}_{N_{t}}}^{w}h_{Ch}^{w}} \right.||^{2}}\end{bmatrix}\quad$

from high to low can result in more than one (up to U_(r)) uniqueordering combination, which satisfy Equations (64) and (65). In thiscase, the next step is to optimize the sum rate using an “adjusted”waterfilling strategy for each ordering combination, and to select theordering combination that offers the largest sum rate.

Alternatively, if Equations (64) and (65) cannot be satisfied for anyordering of

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{t},o}}^{w}\end{bmatrix}{\quad,}$

several remedies exist:

Method XI for Relaxing Received SINR Constraint in Method “D”:

For example, the maximum value, |γ₁ _(S ,o) |², which is obtained as

${\left| \gamma_{1_{,o}} \right|^{2}\overset{\Delta}{=}{\max \left\{ {\left| \gamma_{1_{1,o}} \right|^{2},\left| \gamma_{1_{2,o}} \right|^{2},\ldots,\left| \gamma_{1_{{N_{t} - 1},o}} \right|^{2}} \right\}}},$

is removed from Equations (64) and (65), and placed in a set, {rightarrow over (γ)}_(S) _(o) , and its corresponding indices,

is placed in another set,

_(o). These are repeated until both Equations (64) and (65) aresatisfied. The formed set of squared values in {right arrow over(γ)}_(S) _(o) , is then replaced by a zero value, i.e. the correspondinginformation elements that correspond to {right arrow over (γ)}_(S) _(o)are not transmitted.

Method XII for Relaxing Received SINR Constraint in Method “D”:

Another remedy for the case when Equations (64) and (65) cannot besatisfied for any ordering combination of

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{r},o}}^{w}\end{bmatrix},$

is to reduce Equation (64) by a factor which would allow for an orderingcombination to be found which satisfies both Equations (64) and (65). Inother words, instead of accommodating only a few of the terminal nodesas in the previous strategy, this strategy attempts to accommodate allterminal nodes in a fair fashion.Note: When choosing between the two Methods for relaxing the SINRconstraint, one can rely on the variance of {|γ₁ _(1,o) |², |γ₁ _(2,o)|², . . . , |γ₁ _(Nt−1,o) |²}. If the variance is above a pre-specifiedthreshold, then Method XI is selected, otherwise, Method XII isselected.

Another Embodiment of Pre-Weighting Selection in Method “D”:

For the special case of UL MU-MIMO where N_(r) ^(w)=1, for all values of1≦w≦U_(r), i.e. each receiving device (terminal node) has only oneantenna, DL MU-MIMO reduces to DL MU-MISO. In this case, the set ofordered SINR equations in Equation (61b) is written as:

$\begin{matrix}{\eta_{1_{i,o}}^{w} = {\frac{\left| {\left\{ h_{{Ch}_{i,o}} \right\}_{1}\gamma_{1_{i,o}}^{w}} \right|^{2}}{\left. {\frac{1}{v} + \sum_{l = 2}^{N_{t} - i + 1}} \middle| {\left\{ h_{{Ch}_{i,o}} \right\}_{l}\gamma_{l_{i,o}}^{w}} \right|^{2}} \geq \kappa_{1_{i,o}}^{w}}} & (66)\end{matrix}$

Another Selection of the Ordering of

$\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{r}}\end{bmatrix}\mspace{14mu} {{into}\mspace{14mu}\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{t},o}}^{w}\end{bmatrix}}$

in Method “D”

Based on Equation (66) and the Power constraint in Equation (65), theordering of

$\begin{bmatrix}\eta_{1} \\\vdots \\\eta_{N_{t}}\end{bmatrix}\mspace{14mu} {{into}\mspace{14mu}\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{t},o}}^{w}\end{bmatrix}}$

is based on selecting

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

in such a way as to maximize the sum rate.

One possible way for ordering

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{t},o}}^{w}\end{bmatrix}\quad$

and for selecting

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

which satisfies the constraints in Equations (65) and (66), is toexhaustively search for all possible ordering combinations of

${\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}$

until at least one ordering combination is found which satisfiesEquations (65) and (66). If one or more ordering combinations are foundwhich satisfy Equations (65) and (66), then the next step is to optimizethe sum rate using an “adjusted” waterfilling strategy, and to selectthe ordering combination, which offers the largest sum rate.

Another possible way for ordering

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{t},o}}^{w}\end{bmatrix}{\quad\quad}$

and for selecting

${{\overset{\rightarrow}{\gamma}}_{o}\overset{\Delta}{=}\begin{bmatrix}\gamma_{1_{1,o}} \\\vdots \\\gamma_{1_{N_{t},o}}\end{bmatrix}},$

which satisfies the constraints in Equations (65) and (66), is to sort

$\begin{bmatrix}\frac{\kappa_{1}}{\left. ||h_{Ch}^{w} \right.||^{2}} \\\vdots \\\frac{{}_{}^{}{}_{}^{}}{\left. ||h_{Ch}^{w} \right.||^{2}}\end{bmatrix}\quad$

from high to low for 1≦w≦U_(r). Given that there are U_(r) receivingdevices, sorting

$\begin{bmatrix}\frac{\kappa_{1}}{\left. ||h_{Ch}^{w} \right.||^{2}} \\\vdots \\\frac{{}_{}^{}{}_{}^{}}{\left. ||h_{Ch}^{w} \right.||^{2}}\end{bmatrix}\quad$

from high to low can result in more than one (up to U_(r)) uniqueordering which satisfy Equations (65) and (66). In this case, the nextstep is to optimize the sum rate using an “adjusted” waterfillingstrategy for each ordering, and to select the ordering combination thatoffers the largest sum rate.

Method XIII for Relaxing Received SINR Constraint in Method “D”:

Alternatively, if Equations (65) and (66) cannot be satisfied for anyordering of

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{t},o}}^{w}\end{bmatrix}{\quad,}$

several remedies exist. For example, the maximum value, |γ₁ _(S ,o) |²,which is obtained as

${\left| \gamma_{1_{,o}} \right|^{2}\overset{\Delta}{=}{\max \left\{ {\left| \gamma_{1_{1,o}} \right|^{2},\left| \gamma_{1_{2,o}} \right|^{2},\ldots,\left| \gamma_{1_{{N_{t} - 1},o}} \right|^{2}} \right\}}},$

is removed from Equations (65) and (66), and placed in a set, {rightarrow over (γ)}_(S) _(o) , and its corresponding indices,

is placed in another set,

_(o). These are repeated until both Equations (65) and (66) aresatisfied. The formed set of squared values in {right arrow over(γ)}_(S) _(o) , is then replaced by a zero value, i.e. the correspondinginformation elements that correspond to {right arrow over (γ)}_(S) _(o)are not transmitted.

Method XIV for Relaxing Received SINR Constraint in Method “D”:

Another remedy for the case when Equations (65) and (66) cannot besatisfied for any ordering of

$\begin{bmatrix}\eta_{1_{1,o}}^{w} \\\vdots \\\eta_{1_{N_{t},o}}^{w}\end{bmatrix}{\quad,}$

is to reduce Equation (66) by a factor which would allow for an orderingcombination to be found which satisfies both Equation (65) and (66). Inother words, instead of accommodating only a few of the terminal nodesas in the previous strategy, this strategy attempts to accommodate allterminal nodes in a fair fashion.

Note: When choosing between the two Methods for relaxing the SINRconstraint, one can rely on the variance of {|γ₁ _(1,o) |², |γ₁ _(2,o)|², . . . , |γ₁ _(Nt−1,o) |²}. If the variance is above a pre-specifiedthreshold, then Method XIII is selected, otherwise, Method XIV isselected.

The foregoing systems and methods and associated devices and modules aresusceptible to many variations. Additionally, for clarity and concision,many descriptions of the systems and methods have been simplified. Forexample, the figures generally illustrate one or few of each type ofdevice, but a communication system may have many of each type of device.Additionally, features of the various embodiments may be combined incombinations that differ from those described above.

As described in this specification, various systems and methods aredescribed as working to optimize particular parameters, functions, oroperations. This use of the term optimize does not necessarily meanoptimize in an abstract theoretical or global sense. Rather, the systemsand methods may work to improve performance using algorithms that areexpected to improve performance in at least many common cases. Forexample, the systems and methods may work to optimize performance judgedby particular functions or criteria. Similar terms like minimize ormaximize are used in a like manner.

Those of skill will appreciate that the various illustrative logicalblocks, modules, units, and algorithm steps described in connection withthe embodiments disclosed herein can often be implemented as electronichardware, computer software, or combinations of both. To clearlyillustrate this interchangeability of hardware and software, variousillustrative components, blocks, modules, and steps have been describedabove generally in terms of their functionality. Whether suchfunctionality is implemented as hardware or software depends upon theparticular constraints imposed on the overall system. Skilled personscan implement the described functionality in varying ways for eachparticular system, but such implementation decisions should not beinterpreted as causing a departure from the scope of the invention. Inaddition, the grouping of functions within a unit, module, block, orstep is for ease of description. Specific functions or steps can bemoved from one unit, module, or block without departing from theinvention.

The various illustrative logical blocks, units, steps and modulesdescribed in connection with the embodiments disclosed herein can beimplemented or performed with a processor, such as a general purposeprocessor, a digital signal processor (DSP), an application specificintegrated circuit (ASIC), a field programmable gate array (FPGA) orother programmable logic device, discrete gate or transistor logic,discrete hardware components, or any combination thereof designed toperform the functions described herein. A general-purpose processor canbe a microprocessor, but in the alternative, the processor can be anyprocessor, controller, microcontroller, or state machine. A processorcan also be implemented as a combination of computing devices, forexample, a combination of a DSP and a microprocessor, a plurality ofmicroprocessors, one or more microprocessors in conjunction with a DSPcore, or any other such configuration.

The steps of any method or algorithm and the processes of any block ormodule described in connection with the embodiments disclosed herein canbe embodied directly in hardware, in a software module executed by aprocessor, or in a combination of the two. A software module can residein RAM memory, flash memory, ROM memory, EPROM memory, EEPROM memory,registers, hard disk, a removable disk, a CD-ROM, or any other form ofstorage medium. An exemplary storage medium can be coupled to theprocessor such that the processor can read information from, and writeinformation to, the storage medium. In the alternative, the storagemedium can be integral to the processor. The processor and the storagemedium can reside in an ASIC. Additionally, device, blocks, or modulesthat are described as coupled may be coupled via intermediary device,blocks, or modules. Similarly, a first device may be described astransmitting data to (or receiving from) a second device when there areintermediary devices that couple the first and second device and alsowhen the first device is unaware of the ultimate destination of thedata.

The above description of the disclosed embodiments is provided to enableany person skilled in the art to make or use the invention. Variousmodifications to these embodiments will be readily apparent to thoseskilled in the art, and the generic principles described herein can beapplied to other embodiments without departing from the spirit or scopeof the invention. Thus, it is to be understood that the description anddrawings presented herein represent particular aspects and embodimentsof the invention and are therefore representative examples of thesubject matter that is broadly contemplated by the present invention. Itis further understood that the scope of the present invention fullyencompasses other embodiments that are, or may become, obvious to thoseskilled in the art and that the scope of the present invention isaccordingly not limited by the descriptions presented herein.

What is claimed is:
 1. A method for receiving uplink communications innetwork node of a communication system, the method comprising:determining an ordering combination and determining pre-weighting valuesassociated with the ordering combination for use by at least oneterminal node, the pre-weighting values being determined based on atransmit power constraint and a minimum performance constraint;providing the pre-weighting values to the at least one terminal node;receiving a signal for each of at least one antenna, each receivedsignal comprising a plurality of transmitted signals from the at leastone terminal node based at least in part on the pre-weighting values;and processing the received signal for each of the at least one antennausing the determined ordering combination and the determinedpre-weighting values.
 2. The method of claim 1, wherein determining theordering combination and the pre- weighting values further includesutilizing a sum-rate maximization function.
 3. The method of claim 1,wherein the minimum performance constraint is based on at least areceived signal-to-interference-plus-noise ratio (SINR).
 4. The methodof claim 1, wherein determining the ordering combination and thepre-weighting values comprises evaluating at least one of a plurality ofpossible ordering combinations.
 5. The method of claim 1, whereindetermining the ordering combination and the pre-weighting valuescomprises: generating a set of possible ordering combinations;determining, for each possible ordering combinations, pre-weightingvalues that satisfy one of the transmit power constraint and the minimumperformance constraint; and identifying each one of the set of possibleordering combinations that have determined pre-weighting values thatsatisfy the other one of the transmit power constraint and the minimumperformance constraint.
 6. The method of claim 5, wherein determiningthe ordering combination and the pre-weighting values further comprises:for each of the identified possible ordering combinations, adjusting thecorresponding pre-weighting values based on a sum-rate maximizationfunction; and selecting one of the identified possible orderingcombinations based on a predicted sum-rate associated with each of theidentified possible ordering combinations.
 7. The method of claim 5,wherein, in the identifying step, none of the set of possible orderingcombinations have determined pre-weighting values that satisfy the otherone of the transmit power constraint and the minimum performanceconstraint.
 8. The method of claim 7, wherein determining the orderingcombination and the pre-weighting values further comprises: adjustingthe minimum performance constraint; and repeating the determining stepand the identifying step.
 9. The method of claim 7, wherein determiningthe ordering combination and the pre-weighting values further comprises:setting, for each one of the set of possible ordering combinations, atleast one pre-weighting value associated with the possible orderingcombination to zero; and repeating the determining step and theidentifying step.
 10. The method of claim 1, wherein determining theordering combination and the pre-weighting values comprises: generatinga set of possible ordering combinations, and for each one of the set ofpossible ordering combinations: determining pre-weighting values thatsatisfy one of the transmit power constraint and the minimum performanceconstraint, and evaluating whether the determined pre-weighting valuesassociated with the respective possible ordering combination satisfy theother one of the transmit power constraint and the minimum performanceconstraint.
 11. The method of claim 10, further comprising, for each oneof the set of possible ordering combinations: in the case that thedetermined pre-weighting values associated with the respective possibleordering combination do not satisfy the other one of the transmit powerconstraint and the minimum performance constraint, calculating apre-weighting metric based on the determined pre-weighting values, andin the case that the pre-weighting metric is at or above a threshold,setting at least one of the pre-weighting values associated with thepossible ordering combination to zero and repeating the evaluating step,and in the case that the pre-weighting metric is below a threshold,adjusting all of the pre-weighting values associated with the possibleordering combination and repeating the evaluating step.
 12. The methodof claim 1, wherein processing the received signal for each of the atleast one antenna using the determined ordering combination comprisesperforming successive interference cancelation on the received signalfor each of the at least one antenna in accordance with the determinedordering combination.
 13. The method of claim 1, wherein the at leastone antenna is provided in the network node.
 14. An access node,comprising: a transceiver module configured to communicate with at leastone terminal node including receiving a signal via each of at least oneantenna, each received signal comprising a plurality of signalstransmitted from the at least one terminal node based at least in parton pre-weighting values; and a processor module coupled to thetransceiver module and configured to determine an ordering combinationfor use in processing at least one of the received signals and determinepre-weighting values associated with the ordering combination for use bythe at least one terminal node, the pre-weighting values beingdetermined based on a transmit power constraint and a minimumperformance constraint; provide the pre-weighting values to thetransceiver module for communication to the at least one terminal node;and process the at least one of the received signals using thedetermined ordering combination and the determined pre-weighting values.15. The access node of claim 14, wherein determination of the orderingcombination and the pre-weighting values by the processor modulecomprises utilizing a sum-rate maximization function.
 16. The accessnode of claim 14, wherein the minimum performance constraint is based onat least a received signal-to-interference-plus-noise ratio.
 17. Theaccess node of claim 14, wherein determination of the orderingcombination and the pre-weighting values by the processor modulecomprises evaluating at least one of a plurality of possible orderingcombinations.
 18. The access node of claim 14, wherein determination ofthe ordering combination and the pre-weighting values by the processormodule comprises: generating a set of possible ordering combinations;determining, for each possible ordering combinations, pre-weightingvalues that satisfy one of the transmit power constraint and the minimumperformance constraint; and identifying each one of the set of possibleordering combinations that have determined pre-weighting values thatsatisfy the other one of the transmit power constraint and the minimumperformance constraint.
 19. The access node of claim 18, whereindetermination of the ordering combination and the pre-weighting valuesby the processor module further comprises: for each of the identifiedpossible ordering combinations, adjusting the correspondingpre-weighting values based on a sum-rate maximization function; andselecting one of the identified possible ordering combinations based ona predicted sum-rate associated with each of the identified possibleordering combinations.
 20. The access node of claim 18, wherein, in theidentifying step, none of the set of possible ordering combinations havedetermined pre-weighting values that satisfy the other one of thetransmit power constraint and the minimum performance constraint. 21.The access node of claim 20, wherein determination of the orderingcombination and the pre-weighting values by the processor module furthercomprises: adjusting the minimum performance constraint; and repeatingthe determining step and the identifying step.
 22. The access node ofclaim 20, wherein determination of the ordering combination and thepre-weighting values by the processor module further comprises: setting,for each one of the set of possible ordering combinations, at least onepre-weighting value associated with the possible ordering combination tozero; and repeating the determining step and the identifying step. 23.The access node of claim 14, wherein determination of the orderingcombination and the pre-weighting values by the processor modulecomprises: generating a set of possible ordering combinations, and foreach one of the set of possible ordering combinations: determiningpre-weighting values that satisfy one of the transmit power constraintand the minimum performance constraint, and evaluating whether thedetermined pre-weighting values associated with the respective possibleordering combination satisfy the other one of the transmit powerconstraint and the minimum performance constraint.
 24. The access nodeof claim 23, wherein the processor module is further configure to, foreach one of the set of possible ordering combinations, in the case thatthe determined pre-weighting values associated with the respectivepossible ordering combination do not satisfy the other one of thetransmit power constraint and the minimum performance constraint,calculate a pre-weighting metric based on the determined pre-weightingvalues, and in the case that the pre-weighting metric is at or above athreshold, set at least one of the pre-weighting values associated withthe possible ordering combination to zero and repeat the evaluatingstep, and in the case that the pre-weighting metric is below athreshold, adjust all of the pre-weighting values associated with thepossible ordering combination and repeat the evaluating step.
 25. Theaccess node of claim 14, wherein processing of the at least one of thereceived signals using the determined ordering combination by theprocessor module comprises performing successive interferencecancelation on the at least one received signal in accordance with thedetermined ordering combination.
 26. The access node of claim 14,further comprising the at least one antenna.